Braids, Knots and Applications Patrick D. Bangert School of Engineering and Science International University Bremen Campus Ring 1, 28759 Bremen, Germany. Email: p.bangert@iu-bremen.de Internet: http://www.knot-theory.org Fig. 1. Shells which display a clearly braided pattern (found by the author in Cetraro, Italy). Summary. We introduce braids via their historical roots and uses, make connections with knot theory and present the mathematical theory of braids through the braid group. Several basic mathematical properties of braids are explored and equivalence problems under several conditions defined and partly solved. The connection with knots is spelled out in detail and translation methods are presented. Finally a number of applications of braid theory are given. The presentation is pedagogical and principally aimed at interested readers from different fields of mathematics and natural science. The discussions are as self-contained as can be expected within the space limits and require very little previous mathematical knowledge. Literature references are given throughout to the original papers and to overview sources where more can be learned. A short discussion of the topics presented follows. First, we give a historical overview of the origins of braid and knot theory (1). Topology as a whole is introduced (2.1) and we proceed to present braids in connection with knots (2.2), braids as topological objects (2.3), a group structure on braids (2.4) with several presentations (2.5) and two topological invariants arising from the braid group (2.6). Several 2 Patrick D. Bangert properties of braids are then proven (2.7) and some algorithmic problems presented (2.8). Braids in their connection with knots are discussed by first giving a notation for knots (3.1) and then illustrating how to turn a braid into a knot (3.2) for which an example is given (3.3). The problem of turning a knot into a braid is approached in two ways (3.4 and 3.5) and then the a complete invariant for knots is discussed (3.6) by means of an example. The classification of knots is at the center of the theory. This problem can be approached via braid theory in several stages. The word problem is solved in two ways, Garside’s original (4.1) and a novel method (4.2). Then the conjugacy problem is presented through Garside’s original algorithm (4.3) and a new one (4.4). Markov’s theorem allows this to be extended to classify knots but an algorithmic solution is still outstanding as will be discussed (4.5). Another important algorithmic problem, that of finding the shortest equal braid is presented at length (4.6). Knot theory is very appealing for many reasons but especially because it is so useful is many areas of research. We present applications in classical dynamics (5.1), fluid dynamics (5.2), magnetic relaxation (5.3), path integration (5.4), particle physics (5.5), solid state physics (5.6), quantum field theory (5.7) and the biology of DNA (5.8). Finally the great push towards ideal knots is briefly discussed (5.9). Key words: braids, knots, invariants, word problem, conjugacy problem, rewriting systems, fluid dynamics, path integration, quantum field theory, DNA, ideal knots Braids, Knots and Applications 3 1 Physical Knots and Braids: A History and Overview . . . . . 4 2 Braids and the Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 The Topological Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Origin of Braid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Topological Braid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Presentations of the Braid Group . . . . . . . . . . . . . . . . . . . . . . . The Alexander and Jones Polynomials . . . . . . . . . . . . . . . . . . . . . . . . Properties of the Braid Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic Problems in the Braid Groups . . . . . . . . . . . . . . . . . . . . 7 9 13 16 19 21 24 27 3 Braids and Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 3.2 3.3 3.4 3.5 3.6 Notation for knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braids to Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: The Torus Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Knots to Braids I: The Vogel Method . . . . . . . . . . . . . . . . . . . . . . . . . Knots to Braids II: An Axis for the Universal Polyhedron . . . . . . . Peripheral Group Systems of Closed Braids . . . . . . . . . . . . . . . . . . . . 4 Classification of Braids and Knots . . . . . . . . . . . . . . . . . . . . . . . 50 4.1 4.2 4.3 4.4 4.5 4.6 The Word Problem I: Garside’s Solution . . . . . . . . . . . . . . . . . . . . . . The Word Problem II: Rewriting Systems . . . . . . . . . . . . . . . . . . . . . The Conjugacy Problem I: Garside’s Solution . . . . . . . . . . . . . . . . . . The Conjugacy Problem II: Rewriting Systems . . . . . . . . . . . . . . . . . Markov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Minimal Word Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Applications of Braid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Topology as a Conserved Quantity in Classical Dynamics . . . . . . . . Knot Theory in Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braid Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feynman Path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Braids and Particle Physics: Anyons . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Mechanics: The Yang-Baxter Equation . . . . . . . . . . . . . . Anyons to TQFT’s via the Hopf Invariant . . . . . . . . . . . . . . . . . . . . . Tangles and DNA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideal Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 33 33 34 36 44 50 53 58 60 66 70 78 80 82 88 90 93 97 102 107 4 Patrick D. Bangert 1 Physical Knots and Braids: A History and Overview Everyone knows about knots. We use them to tie our shoelaces and to fasten objects together. For mountain climbers, they become a matter of life and death and for some mathematicians they are the source of bread on the table. If you plan to tie a knot into rope and use it, one of the most vital pieces of information is how much pulling force on the rope the knot can hold before unravelling. Sometimes, the (a) (b) rope will rip before the knot comes undone. This is ideal for mounFig. 2. (a) The simplest method of con- tain climbers. On the other hand, structing a braid is to intertwine two if you use a knot to fasten a boat strings by exchanging their endpoints and to the pier, you want it to hold but (b) the result of the simple exchange. also come undone readily when you want it to. For this, there exist knots which become tighter if you pull the string one way and looser if you pull the string another way. There exists enormous lore on the science of tying knots for various practical purposes (see for example [18]) but very little is known scientifically about the physical properties of various knots [116]. One exception is an interesting project investigating the tension required to break a rope with a knot in it. It has been found that the point of greatest stress is the point of highest curvature in the rope. That means that if the knot has a sharper bend in it than another knot, it will cause the rope to break at a lower tension and indeed at the point of greatest curvature. This was an experimental investigation realized with spaghetti and arose from an inquiry of fishing supplies manufacturers. The question was how much force a fishing line can hold after it has become knotted. The answer is that it depends on the knot type but, in general, much less than if it was unknotted [171]. The investigation of which knot types have a certain amount of curvature if physically tied is closely related to the question of what an ideal knot is; a controversial discussion which we shall go into in section 5.9. Possibly the most important difference between physical and mathematical knots is that mathematics requires the string to be closed. That means that after we tie the knot into a rope, we must glue the ends of the rope together and never undo them. The reason for this is that we are about to consider knots identical if we can continuously deform them into each other. If we had a rope with ends, we could untie the knot and thus every knot would be equal to a segment of straight rope. This would give rise to a use- 1 2 Braids, Knots and Applications 5 less theory and not necessitate a book about it. In order to make this chapter fundamentally amusing for the reader we shall demand ropes without ends. 1 2 4 3 (a) (b) Fig. 3. (a) The points 1 and 3 and the points 2 and 4 form pairs which interchange their positions in turn, thus generating a braid and (b) the three points exchange positions with their image points (drawn in dashed circles) in turn. Knots can be used to tightly compact rope in order to form a bulge which may be used as a button. Knotted buttons were introduced, for example, by the samurai in medieval Japan to hold various small objects to their belts which were made of braided rope (this practise began in the Azuchi monoyama era, 1574 – 1602). The braiding patterns, known as kumihomi, invented during that period became, as many things in Japan, a separate art form practised for its own sake and still exists today. In the baroque period in Europe, women’s clothing was trimmed with braids. Many military uniforms in the 19th century (and some present day dress uniforms) had braids as trimming and decoration and in some cases (South African military, for example) trefoil knots were used as buttons. Braids are very popular today. They are used to make rope as it has been noted that a rope with a braided core has a greater breaking tension while also retaining flexibility and elasticity. Belts and decorations are made using braids and many artists make braiding and knot tying their business. The word “braid” itself comes from a Scottish dialect in the region of Edinburgh and means “broad.” As there is a broad river near Edinburgh, this was called the braid burn and the valley through which it runs is called the braid valley. There are many places in present day Edinburgh named after braids: Braid Veterinary Hospital and The Braid Pub, for instance. Braids are represented in sports by James Braid, the famous golf player and course designer, and in music by the punk band Braid from Urbana, Illinois (disbanded in 2000). The “typical” braid is the one we see in people’s hair. It is made of three strands by alternately taking the left or right strand and passing it over the 6 Patrick D. Bangert middle strand. This braid is so common that the word “braid” is almost synonymous with this pattern. It is every mathematician’s aim to generalist a concept which is already known. In this case, we may increase the number of strands and pass the outer left and outer right strands over more than one of the intermediate strands. So, for example, a five string braid can be made by passing the outer strands over two strands instead of one. Another variety could be an eight string braid in which the outer left passes over three strands and the outer right over four. In tenth century England, the braiding centers used different patterns from each other. London and York opted for the normal three strand braid and Durham was partial to the five and eight string varieties described above. The question of how braids are made is interesting in its own right. Suppose that we have n strings which are fixed at one end (we shall call this the top end) on a straight line and hang down vertically. The other ends are free to move in a horizontal plane P (the bottom end) below the top end. We further label each string by a number from one to n in order from left to right at the top end. Let the intersection of string i and plane P be labelled i also. We may now discuss the braid construction as a series of moves of the points 1 to n in the plane P relative to each other. Let us begin with two points in P which interchange positions at every move, 3 2 see figure 2 (a). This generates a braid 1 on two strings which looks like figure 2 3 (b). This braid is so simple that it hardly 2 1 merits drawing attention to it but the 1 3 pattern can be generalized almost indef2 1 initely. The natural next step is to con3 2 sider four points arranged in a square 2 1 which interchange across the diagonals in 3 turn, see figure 3 (a). This construction 2 method is identical to taking the outer 1 3 string of a four strand braid and passing it over two and under the string just overcrossed. We alternate between using the Fig. 4. The braid which results left and the right outer string for doing from the motion described in figure 3 (b) with the relative positions this. This braid can be rapidly made in of the points at the bottom of each practise and was used widely in prehistoric Denmark to braid leather thongs. horizontal section. Another way to generalist the scheme of figure 2 (a) is to introduce image points, see figure 3 (b). The image points differ from real points in that there are no strings attached to them. We exchange point and image point in numerical order. The way of constructing a braid shown in figure 3 (b) is one in which each string moves to a point Braids, Knots and Applications 7 which is its reflected image across the line joining the other two points. This is how the configuration naturally embeds itself into an equilateral triangle. It is not easy to see what the resultant braid for the construction in figure 3 (b) is so we have drawn it together with the relative positions of the points in figure 4. Note that each exchange of a point with its image point in this scheme generates two crossings in the braid. What is interesting in this example is that we require six exchanges before the points in the plane return to their original relative positions while the braid pattern repeats itself after only two exchanges. The three dimensional structure is thus simpler than the two dimensional dynamical system which gives rise to it. This is an interesting property which can be exploited to classify fundamentally distinct motions in dynamical systems. It should be mentioned that the braid construction of figure 3 (b) is the most optimal way to stir a dye into a liquid [43]. The three points would be rods or paddles of some kind which would be submerged in the liquid and follow the motion prescribed. If one were to record their positions over time then one would obtain figure 4 where the time axis runs vertically upwards. Constructing braids by exchanging points in a plane as we have done above for a few examples can be generalized almost indefinitely. Although we will not undertake this particular generalization, we will develop a number of other general properties of braids. Much more could be said about making braids from real string and the purposes they were made for. 2 Braids and the Braid Group 2.1 The Topological Idea Topology is a branch of mathematics that studies the shape of objects independent of their size or position. If we can deform one object into another by a continuous transformation, then we shall call these objects topologically equivalent. In everyday terms this means that we may bend, stretch, dent, smoothen, move, blow up or deflate an object but we are not allowed to cut or to glue anything. An example of what we may do is shown in figure 5 in which it is shown by an explicit deformation that one can get from a linked structure to an unlinked structure. It is perhaps counter-intuitive that these two objects be considered the same but it is the beauty of topology that we can neglect the particular details of the embedding of the object into space and we must only keep track of certain important global properties. Take for example the doughnut (also called the torus) and the sphere. These two objects are topologically inequivalent. This can be seen easily by observing that the doughnut has a hole while the sphere does not. We can only get rid of the hole by a discontinuous transformation, i.e. in the process of transforming the doughnut into the sphere there will be an instant at which 8 Patrick D. Bangert (1) (2) (4) (3) (5) Fig. 5. This sequence of pictures shows how an initially linked structure (1) is slowly transformed into an unlinked structure (5) by a continuous transformation three stages of which are shown. the hole disappears. This is not allowed in topology and so we have motivated that there are topologically different objects. It is a popular joke that topologists have chipped teeth because they mistake coffeecups for doughnuts. To see that these two are topologically equivalent observe that a coffeecup has a handle. Now imagine making the floor of the cup thicker until the cup can no longer hold any liquid. Then distribute the thick part of the cup evenly throughout the cup, it is easy to see that the result is a doughnut. In the following we shall be concerned with braids and knots. It will be made clear what exactly these are in a moment but the idea is very much the one we are familiar with from everyday use as motivated above by the physical construction of braids. It will be useful to have a few bits of string handy throughout the reading of the following sections so that you may try out some of the results to be presented. This is one of the beauties of knot theory; one can really interact with it. Note that mathematical knots differ from ordinary knots in only one important point: After the knot is tied, the ends of the string are to be glued together and only then is the object regarded as a topological object. If the knot had free ends, all knots would be equivalent as all knots, having been tied, can be untied. Braids, Knots and Applications 9 2.2 The Origin of Braid Theory Few areas of research can trace their origins as precisely as braid theory. Braid theory, as a mathematical discipline, be(1) gan in 1925 when Emil Artin published his Theorie der Zöpfe (2) [16]. A few problems in this first paper were quickly corrected [17] and the study was made algebraic soon thereafter [41]. As (3) we shall see throughout this chapter, braids are closely related to knots and we need to look at Fig. 6. The Reidemeister moves. knots to appreciate braids fully. Knot theory was started in the 1860’s by Peter Guthrie Tait, a Scottish mathematician, who endeavored to make a list of topologically distinct knots in response to a request by William Thompson (later Lord Kelvin) who thought that knotted vortex tubes in the luminiferous ether would make a good model for the elusive atomic theory. This physical application was abandoned when it became clear the the ether did not exist through the Michelson-Morley experiment in 1887. In spite of this, the mathematics was here to stay. Tait first published his work in 1877 at which time he was able to present a long list of knots. It was his purpose to construct the list in order of increasing number of crossings in the knot. How can we tell if a knot is the same as another? Reidemeister has provided us with a convenient way to tell. He proved that the moves in figure 6 are sufficient to get from any diagram of a knot to any equivalent diagram [173]. Reidemeister’s moves are extremely simple, it is almost obvious that they are sufficient (a) (b) to move between any two equivalent diagrams. Actually producing a sequence of Fig. 7. The (a) trefoil knot and (b) moves for two particular diagrams how- its mirror image. ever, is a non-trivial task. As a result, the moves have found their main use in proofs that certain quantities are identical for all diagrams of a particular knot. We call such quantities invariants and they can be integers, real numbers, polynomials, groups, manifolds and other mathematical objects. For example the number of components or closed loops of a knot is an integer invariant as the Reidemeister moves do not perform surgery (cutting or gluing). (0) or 10 Patrick D. Bangert Exercise 2.1. Convince yourself that any topological equivalence move possible on a knot can be reduced to Reidemeister moves. There is only one knot of no crossings – a simple loop of rope – which is called the unknot (recall that a mathematical knot exists on rope without ends). No knot of one crossing can exist as this would simply be a twist of one end of the unknot and can just as easily be undone. This twist is the Reidemeister move zero in reverse, see figure 6. The same applies to any knot of two crossings. Matters become more challenging with three crossings because we hit the trefoil knot. The trefoil knot (see figure 7 (a)) is the knot we usually tie into a shoelace before tying a bow on top of it. Is the trefoil knot the only knot of three crossings? We investigate this question by drawing three double points in the plane (do not differentiate between over and undercrossings yet) which are the intersections of two short line segments each. Then connect the endpoints of the line segments in all possible ways without causing further crossings in the plane. You will find that all of these will unravel to give the unknot by Reidemeister moves of type zero except the trefoil knot. The trefoil knot comes in two natural flavours: the standard type (figure 7 (a)) and the mirror image of the standard type (figure 7 (b)). The mirror image of a knot is obtained by switching all of its crossings (see figure 7 (b)). This method is essentially the one which Rev. Kirkman used in the 19th century to construct a list of all knots up to and including ten crossings. The labour involved in this task is prodigious. To complete the list of all knots of three crossings we must ask: Exercise 2.2. Is the trefoil equivalent to its mirror image? [Hint: This means that you have to find a continuous deformation of the trefoil into its mirror image if they are equivalent or a proof that it is not possible otherwise. The typical method is to look for an invariant if you suspect that they are not equivalent. If you can show that the value of the invariant is different for the two knots, then you have shown that they are different.] The answer to exercise 2.2 is that the trefoil is not equivalent to its mirror image. This is shown by computing an invariant quantity called Alexander polynomial for both knots. We will compute the polynomials for both trefoils in section 2.5. We will motivate the result here by computing a quantity called writhe which is an invariant of all the Reidemeister moves except the zeroth one. Imagine you want to hang a painting on the wall and you are putting a screw into the wall to hold the painting up. You twist the screw clockwise to get it into the wall and counterclockwise when you’ve made a mistake and wish to get it out again. We consider progress positive and mistakes negative so that a crossing in a knot which is achieved by a clockwise rotation of the hands as they follow the orientation of the knot is assigned a weighting of +1; the opposite kind is assigned a weight of −1. The writhe w of a knot is the sum Braids, Knots and Applications 31 41 51 52 61 62 11 63 Fig. 8. The prime knots with fewer than seven crossings and their names from standard tables. The knot 31 is the trefoil knot of figure 7 (a). of the weights over all the crossings. Let us pick the orientation in which the topmost arch on the trefoils in figure 7 points to the left. Then the standard trefoil has w = 3 and the mirror image w = −3. Exercise 2.3. Verify that the trefoil has a writhe of 3 and its mirror image a writhe of -3 if the orientation of the topmost arch is directed to the left in figure 7. Exercise 2.4. Prove that writhe is invariant for all Reidemeister moves except move zero. Using such methods, it is possible to construct a large table of knots. In figure 8 we show the first seven knots after the unknot. It is understood that the two ends of the rope must be joined to yield the mathematical knot. We present them in this fashion for ease of understanding and practical experimentation. 12 Patrick D. Bangert Exercise 2.5. Retie all the knots in figure 8 with string and convince yourself that they are truly distinct and that you can not get a knot of fewer than seven crossings which is not listed. The question is: Can we find a general method to determine equality or otherwise for any two knots? The answer is yes, but with qualifications. There exists a method due to Waldhausen, Hakken, Hemion and others but it is so inefficient that it is not possible to use for knots for which we do not know the answer already [103]. There exists a theorem due to Alexander which states that every knot can be represented by a braid [11] which we prove in theorem 2.11. This gave the motivation for people to study braids in order to try to help classify knots. The greatest thrust came from Markov who proved a result for braids similar to Reidemeister move result for knots [142]. Using Markov’s theorem to classify knots has proven difficult however and the search continues. In the sections to come we shall review the results and present the solutions as far as they are known. Braids have proven tremendously useful in spite of the fact that they have not lead to a complete knot classification scheme. Many invariant of knots are naturally defined on braids. The most revolutionary invariant, the Jones polynomial, was discovered using braid theory. Beyond this, braids have many applications to various fields as we shall discover in the sections to come. An operation to combine knots can be defined which we are going to call knot addition and denote it by #. Definition 2.6. Given two knots K and L, we define the knot sum K#L as the knot obtained by cutting both K and L at a random location and gluing them together with respect to their orientations. This is a simple operation but it is not at all obvious that it is well defined. One can show that: (1) The sum is independent of the points on K and L chosen as cutting points [158], (2) any knot can be uniquely factorized into a finite length sum of knots [178], (3) this sum may actually be determined [179]. Property (1) makes the concept well-defined. The second property establishes the existence of prime knots, i.e. knots that may not be decomposed into the sum of others and also that classifying prime knots will classify all knots. The third property means that this is, at least in theory, possible to actually compute. However, the algorithm to find the unique decomposition is the algorithm alluded to previously and thus this is not a practical method. It can be shown that there does not, in general, exist an inverse to the operation of addition of knots. As one may show that knot addition is associative, knots form a semi-group but not a group under the operation of addition [158]. As we shall find braids to have a group structure and that any knot can be represented as a braid, braid theory may seem the way to go in order to classify knots. Much work in this direction has been done and many powerful results obtained from such a procedure. Braids, Knots and Applications 13 2.3 The Topological Braid l1 a1 a2 a3 A A C l2 b1 b2 (a) b3 C B B (b) Fig. 9. (a) An example of the definition of a topological braid (see definition 2.7) and (b) an elementary deformation as defined in definition 2.8. We know from section 1 what a braid is. Mathematically, we have to be slightly more careful. Definition 2.7 (n-braid). Let l1 and l2 be two parallel lines in a plane P and let A = {a1 , a2 , · · · , an } and B = {b1 , b2 , · · · , bn } be sets of points on l1 and l2 respectively. An n-braid is a set of (possibly oriented) n nonintersecting polygonal curves which have exactly one endpoint in A and one in B such that all points in A or B are the endpoints of exactly one of these curves and such that any line l parallel to l1 and l2 crosses any curve in at most one point. An example of a 3-braid, in which we have labelled the lines l1 and l2 as well as the point sets A and B, is shown in figure 9 (a); note that the plane P is understood to be the plane of the paper. In this example, we have 3 curves each of which have two endpoints, one in A and one in B. These curves go from l1 to l2 monotonically, they do not double back on themselves. This is the meaning that no line parallel to l1 and l2 may cross any curve in more than one point. In fact it is this requirement that makes braids substantially simpler than knots and allows a group structure to be defined on braids. An n-braid in the form of definition 2.7 is also called an open braid. We shall drop the n from n-braid when no confusion can arise. The normal braid which is braided into peoples’ hair fulfills these requirements. One end of all the strings is fixed on the person’s head and others are held in place by some form of rubber band. The braiding in the middle is done in a way that each bundle of hairs goes from top to bottom monotonically. Some Celtic designs used as borders in the Book of Kells or other illuminated books or more commonly used as trimmings for medieval clothing, necklaces, pendants and belts are not usually braids conforming to this definition as their strings often return to a point close to their origin and thus contain a local maximum or minimum. 14 Patrick D. Bangert Whenever we define a new mathematical object, we desire an equivalence relation for the possible instances of this object. As braid theory is a topological pursuit, we will allow ourselves the usual freedom of topology which means that we will allow the object to be distorted in any way as long as this can be done continuously. So we may bend, stretch and pull a string but we may never cut a string or glue two strings together; such actions are called surgery and are said to change the topology. For braids, it is clear that if we do not fix the endpoints of the curves, we shall be able to transform any braid into any other yielding a rather boring theory; thus we also require the ends to be fixed. We will call the equivalence relation for braids under these conditions isotopy. Before we can define isotopy, we must define what we mean by a topological deformation. An elementary deformation is the basis for all topological deformations. Definition 2.8 (elementary deformation). Suppose that a braid string (recall that it was defined as a polygonal curve) has points A and B as vertices. We may then create a further point C, delete the segment AB and create the segments AC and CB. This deformation, and its inverse, is called an elementary deformation if and only if the triangle ABC does not intersect any other strings and only meets the current string along its side AB. See figure 9 (b) for an illustration. Definition 2.9 (braid isotopy). Two braids α and β are called isotopic, denoted α ≈ β, if and only if α can be transformed into β using a finite number of elementary deformations. Suppose we were to label the string which intersects the point bi by i. On l2 , the string labels from left to right would thus be in numerical order whereas on l1 they may not be ordered numerically. If we list the numerical labels of the strings which intersect l1 from left to right, we obtain a permutation on the set of integers {1, 2, · · · , n}. A braid thus induces a permutation on the set of the first n integers. For example, the braid in figure 9 induces the permutation [2, 3, 1]. The fact that the induced permutation is an equivalence class invariant follows immediately from the requirement that the endpoints be fixed. An open braid may be closed to yield a knot. See figure 10 for the closure of the braid from figure 9. A closed braid, denoted α, is obtained from an open n-braid α by deleting l1 and l2 and connecting points ai and bi with non-intersecting polygonal curves in P for all i : 1 ≤ i ≤ n. It is clear that the closure of any braid yields a knot. Thus some knots may be represented as closed braids. Unfortunately determining the closed braid, given a knot, is not so easy. It is however possible and we shall solve this problem in sections 3.4 and 3.5. The proof of the fact that all knots may be represented as closed braids, Alexander’s theorem, gave the initial momentum for studying braids in detail [11]. Artin took up the challenge and constructed a theory of braids with a view to use them to deal with knots. Braids, Knots and Applications a1 a2 a3 b1 b2 b3 15 Fig. 10. The braid from figure 9 is closed here. It is immediate upon simple transformation that this knot is the same as the simple loop or the unknot. Exercise 2.10. A simple knot invariant is the number of components a knot has. Show that the number of components of the knot α is equal to the number of cycles in the permutation that the braid α induces. Alexander’s theorem is usually proved by giving a topological method with which to deform a knot into a closed braid. There exist several distinct methods of doing so but most are not suitable for use; they are only employed to establish the theorem. The proof given here is fundamentally different and new to the best knowledge of the author. The theorem assumes that the knot is oriented but if not the transformation is accurate up to orientation change. Theorem 2.11 (Alexander [11]). Every knot may be represented as a closed braid. Proof. Consider an oriented straight line a in R3 which we will call the axis. Choose a point O on a and construct a cylindrical polar coordinate system which has O as its origin. The positive z, or upward vertical, direction is directed parallel to a in the direction of its orientation. The polar angle φ increases in the counterclockwise direction, as usual. Using this system, the theorem claims that every knot K can be deformed with respect to a in such a way that the polar angle of a point P going along any component of K strictly increases or more simply: As we travel along the knot we will go around the axis a without ever changing our counterclockwise direction. Suppose we have n straight line segments si for 1 ≤ i ≤ n with endpoints Ri and Si such that the polar angle of Si , φ(Si ) is is larger than φ(Ri ). We may form any knot by subdividing these segments into a finite number of straight subsegments, moving the endpoints of the subsegments and performing surgery which identifies Ri and Si for all i. Here, we will form the knot K by keeping Ri fixed and moving the point Si creating new points Qi,j indexed by j as necessary. Whenever it becomes necessary to move Si to a 16 Patrick D. Bangert position of lower polar angle than the last Qi,j created, move Si once around a creating a suitable number of points doing so and then continuing. After the required knot is formed, we perform the surgery of identifying Ri and Si . By definition of a knot as a polygonal curve such a construction is always possible. Q6 Q5 O R Q13 Q4 z Q9 f S S Q2 R R Q12 Q3 Q2 S Q1 Q7 Q8 O Q14 O Q1 Q10 Q11 (a) (b) (c) Fig. 11. Constructing the trefoil knot as a closed braid, see proof of theorem 2.11 for a discussion. An example of this method applied to forming the trefoil knot is given in figure 11. This proves the theorem. ⊓ ⊔ 2.4 The Braid Group Fig. 12. The generator σi and its inverse σi−1 for the braid group Bn . Braids, Knots and Applications 17 We note that any braid can be represented by a vertical stack of two types of crossing, see figure 12. When all strings are vertical apart from strings i and i + 1, we will denote this crossing by σi or σi−1 depending on whether string i overcrosses or undercrosses string i + 1 respectively. It is thus clear that any braid can be specified by a string of these symbols. We agree to the convention that the left to right direction of the symbols representing a braid shall correspond to the upward direction of the braid; that is, the lowest crossing corresponds to the first symbol. This is a convention and some other authors use the opposite convention. While care is required, no serious consequences arise from this choice. For example, the braid in figure 2 (b) is σ15 (the power means that the symbol σ1 was repeated five times) 3
and the braid in figure 4 is σ1 σ2−1 σ1−1 σ2 . From now on, we shall denote a braid by these symbols. Definition 2.12 (braid word). We will call any sequence of σi±1 a braid word. Definition 2.13 (positive and negative braid word). If a braid word contains only σi (and no σi−1 ) then it will be called positive. However, if it is contains only σi−1 (and no σi ) then it will be called negative. 5cm (a) (b) (a) (1) (b) (2) Fig. 13. (1) Two non-interfering crossings can be listed in either order and (2) a crossing may be moved underneath an arch which overcrosses both strings involved Consider the braid σ3 σ1 displayed in figure 13 (1a). This braid is clearly topological equivalent to the braid σ1 σ3 displayed in figure 13 (1b). More generally, every time two neighboring crossings are on distinct pairs of strings, the order in which these crossings are listed in the braid word does not topologically matter. Thus we arrive at the rule that σi σj ≈ σj σi for |i − j| > 1 (1) 18 Patrick D. Bangert which is usually called the far commutation relation as it embodies the fact that generators sufficiently far from each other commute. It also becomes clear that if we have an arch which over or undercrosses two strings which then cross (see figure 13 (2a)), this crossing may be moved onto the other side of the arch (see figure 13 (2b)). Thus the braids σ1 σ2 σ1 and σ2 σ1 σ2 are topologically equivalent. As this can hold anywhere in a braid, we arrive at the second rule that σi σi+1 σi ≈ σi+1 σi σi+1 (2) which is typically called the braid relation. We find this name too vague and so we will refer to this relation as the bridge relation as it symbolizes that anything not in conflict with the principal pillars may move freely both below and above a bridge. After some experimentation, one notices that all the moves one may make on a braid while preserving its topology can be reduced to applying the rules in equations 1 and 2 to the braid word. We would like to prove that this is always so. Theorem 2.14. (Artin [16] [17]) The equivalence relation upon braid words defined by the relations 1 and 2 is identical to the equivalence relation of braid isotopy (see definition 2.9) upon the braids represented by the braid words. Proof. Recall that any knot may be represented as a closed braid. The braid contains all the crossings and Reidemeister’s moves define equivalence of knots. Thus braid equivalence is Reidemeister equivalence of the braid. Move zero would create an object which is not a braid but the others apply. Translating these into the σi notation and simplifying yields the given presentation. From figure 12, it is clear that σi−1 is the inverse of σi . As we represent a braid by a braid word in the σi from the bottom up, we can easily concatenate two braids together. The braid αβ is constructed from the braids α and β by identifying the top ends of α with the bottom ends of β. It is obvious that concatenation is associative, i.e. (αβ)γ ≈ α(βγ) and that the concatenation of two braids is a braid. Suppose ι is the braid of n vertical strings and no crossings. We have ια ≈ αι ≈ α for any n-braid α. The braid ι acts as an identity. Since we have closure, associativity, inverses and an identity, the set of n-braids forms a group generated by the generators σi . We will denote this group by Bn and refer to this family of groups as the braid groups. Because of theorem 2.14, Bn has the presentation   σi σi+1 σi ≈ σi+1 σi σi+1 , Bn = {σ1 , σ2 , · · · , σn−1 } : (3) σi σj ≈ σj σi for |i − j| > 1 It is instructive to consider the braid group from another point of view which curiously leads to the same presentation as given above in equation 3. Consider the space between the two parallel planes which contain l1 and Braids, Knots and Applications 19 l2 respectively after the braid has been removed from it. This space has a fundamental group which may be represented by a series of loops beginning and ending on some randomly chosen base point b and going around the removed braid strings. There are n distinct such loops which we shall call xi with i running from 1 to n. The fundamental group is free of rank n with the xi as generators. x1 is the loop around the first string from the left on each level and so on for the other xi . From level to level a reassignment of generators becomes necessary. This is called an automorphism and the particular one we need here, ai is defined by ai : xi → xi+1 ; xi+1 → x−1 i+1 xi xi+1 ; xp → xp (p = i, i + 1) (4) The map α : σi → ai is a homomorphism of Bn into the automorphism group of Fn , the free group of rank n. It can be shown that the ai generate a group with presentation identical to equation 3 under the homomorphism α. In other words, a braid word may be regarded as an automorphism of Fn . As the mapping merely consists of a change of symbol for the generators, it is frequently useful not to make a distinction between a braid word as an element of Bn and as an automorphism of Fn . In the next section, we will introduce some other presentations of the braid group. 2.5 Other Presentations of the Braid Group The presentation of Bn given in equation 3 is called the Artin presentation as it was Artin who first used it in the paper which founded the field. The fact that braids admit a group structure simplifies their treatment tremendously. It can be shown that knots do not admit a group structure and this is one reason why the problem of deciding if two knots are equal is so different from the similar question about braids. It is difficult, however to extract useful information from a presentation of a group. For this reason it is useful to search for other presentations of the same group with special properties. The Artin presentation has the appeal that it is very topological. It is easy to draw the braid given the braid word and it is easy to read off the braid word from a braid. Furthermore, both the far commutation and the bridge relations are simple to perform. One disadvantage is that each braid group has a different number of generators. Consider putting a = σ1 σ2 · · · σn−1 and σ = σ1 . After some manipulation, we find the presentation  n (5) Bn = {a, σ} : an ≈ (aσ)n−1 , σa−j σaj ≈ a−j σaj σ for 2 ≤ j ≤ 2 which we call the Coxeter presentation. The Coxeter presentation has the advantage that all braid groups have just two generators but we have lost some of the topological correspondence. It would be nice to have a matrix representation of the braid groups. To this end, we will represent the identity matrix of n rows and columns by 20 Patrick D. Bangert In (recall that the identity matrix has unity entries in the leading diagonal and zeros everywhere else). Then we define the mapping φn (σi ) of an Artin generator σi to a matrix of n rows and n columns whose entries are Laurent polynomials in the variable t (Laurent polynomials allow both positive and negative powers of the variable). Expressed formally, this means
  φn : Bn → GL n, Z t±1 (6) where Z is the ring of polynomials. The mapping is defined by   Ii−1 0 0 0  0 1−t t 0   φn (σi ) =   0 1 0 0  0 0 0 In−i−1 (7) It can easily be shown that φn is a homomorphism,
  i.e. that φn (σi σj ) = φn (σi )φn (σj ) where multiplication in GL n, Z t±1 is the usual matrix multiplication. A representation is termed faithful when the mapping φ giving rise to it is injective, i.e. when x = y implies φ(x) = φ(y). This representation is faithful for n ≤ 3 [139] and not for n ≥ 5 [29]. For n = 4 the answer is unknown. It should now be easy to write down a matrix representation for any braid word. For example, if n = 3, then for φn (σ1 σ2 ) we have      1−t t 0 1 0 0 1 − t t − t 2 t2 0 0  (8) φn (σ1 )φn (σ2 ) =  1 0 0   0 1 − t t  =  1 0 1 0 0 01 0 1 0 This representation is called the Burau representation. Recently a new presentation was invented by Birman, Ko and Lee [31] which they used to solve the word and conjugacy problem in a new way. The generators akl are defined by
−1 −1  −1 akl = (σk−1 σk−2 · · · σl+1 ) σl σl+1 σl+2 · · · σk−1 (9) Topologically this is a crossing between two arbitrary braid strings k and l. In akl , string k overcrosses string l and both strings overcross all other string in between them to be able to cross and then overcross the in between strings again to return to their original (but now switched) positions. Using these generators, Bn has the presentation Bn =  ats asr = atr ats = asr atr , {ats ; n ≥ t > s ≥ 1} : ats arq = arq ats for (t − r)(t − q)(s − r)(s − q) > 0  (10) which we call the band-generator presentation. In the Artin representation, we number the strings from left to right on each level. Suppose we were to Braids, Knots and Applications 21 label each string with a unique label which it would carry throughout the braid. At the bottom of the braid, we number the strings from one to n as we go from left to right. Each crossing in which string i overcrosses string j is labelled with the generator gij . This presentation is called the colored braid presentation as the string labels act like each string was made of a separate color. This representation retains the complete information of the braid but it is not immediately apparent whether a crossing is positive or negative in the Artin sense. Note that this feature means that the generators gij are self 2 inverse, gij ≈ e the identity. We easily write down the braid group relations in this presentation to get Bn =  gij gik gjk = gjk gik gij , 1 ≤ i, j ≤ n : gij gkl = gkl gij for {gij } for i = j (i − k)(i − l)(j − k)(j − l) > 0  (11) We note that in the colored braid presentation, the fundamental braid ∆n takes the form ∆n = g12 g13 · · · g1n g23 g24 · · · gn−1 n (12) gij (13) = n−1  n  i=1 j=i+1 2.6 The Alexander and Jones Polynomials The Burau representation of the braid group is important in the definition of a revolutionary knot-invariant called the Alexander polynomial. We begin with a braid word α, construct its Burau representation φn (α) and take the determinant of the matrix [φn (α) − In ]1,1 where the subscript indicates that the first row and column should be deleted. This determinant can be shown to be a topological invariant of the closure of the braid α and is denoted by △α such that (14) △α (t) = det[φn (α) − In ]1,1 Thus the Alexander polynomial of the braid σ1 σ2 of the above example is △ = 1 which happens to be the same value as the Alexander polynomial for the unknot. While σ1 σ2 is actually isotopic to the unknot, we could not conclude this from its Alexander polynomial. The Alexander polynomial is an incomplete invariant in that we can only say that if △K = △K ′ , then K = K ′ . The converse is not true, in general. Nevertheless, the Alexander polynomial is very important in knot theory. Let us compute the Alexander polynomial for both trefoils (see figure 7). The ordinary trefoil is the closure of σ13 and its mirror is the closure of σ1−3 . Now we have    
3
−3  (1 − t)3 0 (1 − t)−3 0 φ2 σ 1 = φ2 σ 1 = (15) 0 t3 0 t−3 22 Patrick D. Bangert and thus △σ 3 = t 3 − 1 1 △σ3 = t−3 − 1 1 (16) And since the Alexander polynomials of the two trefoils are distinct, the knots must be distinct. Many other polynomials invariants have been devised after the Alexander polynomial, the most important is the Jones polynomial. The Jones polynomial is also an incomplete invariant of knots which was originally constructed in terms of braids. We shall not prove that it is an invariant, nor that it is incomplete but we shall give an easy method to obtain it. Even though it is incomplete, it is a very powerful invariant in that it distinguishes many knots not distinguished by other invariants such as the Alexander polynomial. We define the tensor product of a p × q matrix A and a r × s matrix B by   a11 B a12 B · · · a1q B a21 B a22 B · · · a2q B    (17) A⊗B = . .. . . ..   .. . .  . ap1 B ap2 B · · · apq B A ⊗ B is clearly a pr × qs matrix. We also define the trace of a n × n matrix C by n  cii (18) tr(C) = i=1 Consider the matrix µ(t) =   10 0t (19) and define µ⊗n = µ ⊗ µ ⊗ · · · ⊗ µ    n times (20) then µ⊗n is a 2n × 2n matrix and tr (µ⊗n ) = (1 + t)n . Furthermore let     1 0 0 0 1 0 0 0 0 0 −t1/2 0 0 1 − t−1 −t−1/2 0 −1  ;  R= R (21) = 1/2 0 −t 0 −t−1/2 1 − t 0 0 0 0 0 0 1 0 0 0 1
2 where we use the convention that t1/2 = t. After all these definitions, we are ready to define a map from the braid 2n × 2n  matrices with Laurent polynomial entries (Φn : group Bn
to the  1/2 n Bn → M 2 , Z t , t−1/2 ) given by Φn (σiǫ ) = I2 ⊗ I2 ⊗ · · · I2 ⊗Rǫ ⊗ I2 ⊗ I2 ⊗ · · · I2       i − 1 times n − i − 1 times (22) Braids, Knots and Applications 23 where ǫ = ±1 and I2 is the 2 × 2 identity matrix. For some general braid word β we have β = σiǫ11 σiǫ22 · · · σiǫkk (1 ≤ i1 , i2 , · · · , in < n; ǫj = ±1)
ǫ1 
ǫ2 
 Φn (β) = Φn σi1 Φn σi2 · · · Φn σiǫkk (24) ǫ(β) = ǫ1 + ǫ2 + · · · + ǫk (25) (23) If we now define then the Jones polynomial is given by Vβ (t) = t ǫ(β)−n+1 2 tr (Φn (β)µ⊗n ) 1+t (26) and we have that if β1 ≈ β2 , then Vβ1 (t) = Vβ2 (t) for any two braids β1 and β2 (not necessarily members of the same braid group). Exercise 2.15. Compute the Jones polynomial for the 2-braid σ1 , the closure of which is clearly the unknot (solution follows). We just compute and obtain   1 0 0√ 0 10 1−2+1 0 0 − t 0 0 t t 2  √ tr  Vσ1 (t) = 1 + t 0 − t 1 − t 0 0 0 00 0 0 0 1 0 0 t2 0  0  0  = 1 0  t3 (27) It is a well-known conjecture that the Jones polynomial recognizes the unknot, that is to say that the Jones polynomial is equal to 1 if and only if the knot is isotopic to the unknot. Many people have been searching for a proof and a counterexample and none has been found thus far. Resolving this question is one of the more important open problems in knot theory. Proving that the Jones polynomials is an invariant of closures of braids is easy. For reasons of space, we simply outline the equations that need to be checked by straightforward calculations. Theorem 2.16. The Jones polynomial is a knot invariant. Proof. By direct calculation, check 
 1. The mapping Φn is a homomorphism from Bn to M 2n , Z t1/2 , t−1/2 . This can be done by checking that Φn (σi σj ) = Φn (σj σi ) ; (|i−j| > 1) (28) 2. By Markov’s theorem, two closed braids are isotopic if and only if they can be reached from each other by conjugacy and stabilization. Thus we need to check if the Jones polynomial is invariant under these two moves, Vγβγ −1 (t) = Vβ (t); Φn (σi σi+1 σi ) = Φn (σi+1 σi σi+1 ) Vβ (t) = Vβσn±1 (t) (γ, β ∈ Bn ) (29) 24 Patrick D. Bangert The polynomial invariants can be calculated using so-called skein relations. These relations are expressions relating the polynomials of knots that are identical except for a single local change. In braid language, the Alexander polynomial of the closed braid β ∈ Bn , Aβ (z) is given by the relation Aβσi (z) − Aβσ−1 (z) = zAβ (z) i (30) together with the boundary condition that if the closure of β is the unknot, A(z) = 1. The Jones polynomial Jβ (z) has the same boundary condition but takes the skein relation   √ 1 Jβσi (z) − tJβσ−1 (z) = z − √ Jβ (z) (31) i z t Using these skein relations, it is possible to calculate the polynomial invariants straight from the diagram by successively eliminating crossings [116]. 2.7 Properties of the Braid Group In this section we will prove various propositions which enumerate the basic properties of braids. The Artin presentation will be used for all these as for most of the rest of the chapter. Proposition 2.17. The braid group Bn has infinite order; it contains an infinite number of distinct braids. Proof. Consider the braid α = σ1 ∈ B2 . Clearly, αi ≈ αj if and only if i = j. Therefore the family of braids αi ∈ B2 is infinite and thus Bn is of infinite order. ⊓ ⊔ Recall that the center of a group is the set of all those elements which commute with all other elements in that group. The centre of the braid groups will become important in many places and so we will construct it. Definition 2.18 (fundamental word). The fundamental braid word ∆n ∈ Bn is defined by ∆n = σ1 σ2 · · · σn−1 σ1 σ2 · · · σn−2 · · · σ1 σ2 σ1 (32) It is a simple matter of applying the braid group relations to find that n ∆2n = (σ1 σ2 · · · σn−1 ) (33) Exercise 2.19. Show that equation 33 follows from equation 32 by using the relations in the Artin presentation of the braid group Bn given in equation 3. Braids, Knots and Applications 25 Proposition 2.20. The center C(Bn ) of the braid group Bn is the set  (34) C(Bn ) = ∆2i n for any (positive, negative or zero) integer i. Proof. Recall that an element is said to be in the center of a group if and only if every element of the group commutes with it. We are then claiming that 2i σj ∆2i (35) n ≈ ∆n σj for all i and j. We are also claiming that any element which commutes with all others has is an even multiple of ∆n . Topologically, it is clear that ∆n commutes with everything by recognizing that ∆2n is a full twist of the entire braid and thus any other braid must commute with it. Algebraically, the proof is also simple (just use the braid group relations to move σj through 2 ∆2i n ) but lengthy and thus will be left to the reader. The proof that ∆n not only is in the center but generates it is complicated and lengthy (a central part is played by computations related to the Reidemeister-Schreier theorem) and so the interested reader is referred to the literature [54]. ⊓ ⊔ Definition 2.21. The ascending braid word aj is defined by aj = σ1 σ2 · · · σj . Proposition 2.22. In Bn , we have σi aj ≈ aj σi−1 for 1 < i ≤ j < n. Proof. By use of the defining relations we have σi aj = σi σ1 σ2 · · · σi−2 σi−1 σi σi+1 · · · σj ≈ σ1 σ2 · · · σi−2 σi σi−1 σi σi+1 · · · σj ≈ σ1 σ2 · · · σi−2 σi−1 σi σi−1 σi+1 · · · σj ≈ σ1 σ2 · · · σi−2 σi−1 σi σi+1 · · · σj σi−1 = aj σi−1 (36) (37) (38) (39) (40) ⊓ ⊔ Proposition 2.23. For all i we have σi ∆n ≈ ∆n σn−i . Proof. First consider the case i = 1. Then, σ1 ∆n = σ1 an−1 σ1 σ2 · · · σn−2 ∆n−2 (41) ≈ an−1 an−2 ∆n−2 σn−1 = ∆n σn−1 (44) (45) ≈ σ1 σ2 σ3 · · · σn−1 an−1 ∆n−2 by prop. 2.22 ≈ an−1 an−2 σn−1 ∆n−2 Now consider the cases 1 < i < n, (42) (43) 26 Patrick D. Bangert σi ∆n = σi an−1 an−2 · · · an−i+1 ∆n−i ≈ an−1 an−2 · · · an−i+1 σ1 ∆n−i by prop. 2.22 ≈ an−1 an−2 · · · an−i+1 ∆n−i σn−i = ∆n σn−i (46) (47) (48) (49) ⊓ ⊔ Definition 2.24. Two positive n-braid words a and b are called positively equal if and only if there exists a sequence of words Wi with 0 ≤ i ≤ p for some finite p for which W0 = a, Wp = b, Wi is obtained from Wi−1 by a single application of one of the defining relations of Bn , and all Wi are positive. In other words, two braids are positively equal if we can transform one into the other without ever having to use an inverse generator. Proposition 2.25. Two equal positive braid words are positively equal. Proof. The statement of the proposition is a trivial corollary to the statement that the mapping ξ : σi → si for 1 ≤ i < n induces an embedding of the semigroup Sn into Bn where Sn is defined by   si si+1 si ≈ si+1 si si+1 , Sn = {s1 , s2 , · · · , sn−1 } : (50) si sj ≈ sj si for |i − j| > 1 To show this statement, by Öre’s theorem [58], it suffices to show that Sn is both right and left cancellative and is right reversible. By right and left cancellative we mean that if ax = ay, then x = y and if xa = ya, then x = y respectively. Right cancellative follows from left cancellative by the fact that x = y implies R(x) = R(y) where R is the operator which reverses the order of the generators in the word. Left cancellative will be shown by induction over the word length l. The result is trivial for l = 0, 1. Suppose the result is true up to l = k, then By right reversible we mean that if x, y ∈ Sn , then there exist u, v ∈ Sn such that ux = vy. Consider ∆n = an−1 an−2 · · · a1 (51) + ≈ an−1 an−2 · · · at+1 (at−1 at−2 · · · a1 ) at = yt σ t (52) (53) where the word x+ is obtained from the word x by increasing the indices of all generators by one. This shows that the fundamental braid is positively equal to a braid ending in any of the generators. This proof can be trivially reversed to show that it is also positively equal to a braid beginning with any of the generators. Putting u = ∆m n where m is the word length of y and making use of proposition 2.23 easily shows that ux = vy. This proves the proposition. ⊓ ⊔ Braids, Knots and Applications 27 2.8 Algorithmic Problems in the Braid Groups We have defined braids, elucidated their connection with knots and found a group structure on braids. This group structure has certain properties of which we enumerated a few important ones in the last section. There are a number of questions which we may readily ask about braids, the most significant of which we shall describe in this section. Clearly, we wish to know both necessary and sufficient conditions for equivalence. Definition 2.26 (word problem). Given two braid words α, β ∈ Bn , the word problem asks whether α ≈ β. The question of whether α ≈ β is identical to the question of whether αβ −1 ≈ e, the identity in Bn . Thus the word problem reduces to recognizing the identity element. Recall that two elements a and b of some group G are called conjugate, denoted a ≈c b, if and only if there exists a c ∈ G such that a ≈ cbc−1 . The conjugacy condition becomes αγ ≈ γβ for two braid words α, β ∈ Bn to be conjugate with respect to a third braid word γ ∈ Bn . We are naturally interested under what conditions such a commutation relation exists. Definition 2.27 (conjugacy problem). Given two braid words α, β ∈ Bn , the conjugacy problem asks whether there exists a third braid word γ ∈ Bn such that αγ ≈ γβ. Suppose α ≈ β, then we also have αγ ≈ γβ for γ ≈ e. Thus any equal braid words are also conjugate but two conjugate braid words are not necessarily equal and so the conjugacy problem subsumes the word problem. The most central question in knot theory is that of classification: Given two knots K1 , K2 are they topologically equivalent K1 ≈ K2 ? Markov has found it possible to translate this question into a question about braids. Imagine closing the braid cbc−1 . We may move the subbraid c through the closure part so that it emerges on top of the rest of the braid, i.e. becomes bc−1 c ≈ b. In other words, conjugation of a braid preserves closed braid isotopy: Any two conjugate braids represent equivalent knots. Conjugation is a sufficient condition but unfortunately not necessary. The braid e ∈ B1 is just a single vertical line segment which closes to the unknot. The braid σ1 ∈ B2 is the braid of single positive crossings which also closes to the unknot. As the braids are in different braid groups, they have a different number of strings and are thus not conjugate to each other. Such braids can be related through a move called stabilization. Definition 2.28 (stabilization move). Given a braid α ∈ Bn , the operation α ↔ ασn±1 is called stabilizing the braid and is referred to as the stabilization move. 28 Patrick D. Bangert Note that stabilization changes the number of strings in the braid by one and that we refer to both the addition and removal of a string as stabilizing. Starting with e ∈ B1 , we stabilize once to obtain σ1 ∈ B2 . In this way, we can move between two braids in different braid groups. Markov’s theorem states that conjugacy and stabilization are enough for knot equivalence. Theorem 2.29 (Markov [142], Birman [30]). Given two braids α ∈ Bn and β ∈ Bm , we have α ≈ β if and only if β may be obtained from α by a finite sequence of conjugacy or stabilization moves; we denote this by α ≈M β and call α and β Markov equivalent. In [142], Markov stated this theorem but the first complete published proof appeared in [30]. We shall not prove this theorem here as it is difficult and lengthy and would distract from the flow of the chapter. It is clear that we are interested in approaching knot classification from a braid theory point of view. Definition 2.30 (Markov problem). Given two braid words α ∈ Bn and β ∈ Bm , the Markov problem or algebraic link problem asks whether α ≈M β. By Markov’s theorem, a solution to the Markov problem would provide a solution to the knot classification problem. As the Markov problem subsumes the conjugacy problem, we are presented with a hierarchy of three combinatorial problems in group theory which would have significant weight if a solution were found. Solutions exist to the word and conjugacy problems and we shall develop new solutions in later sections. A solution to the Markov problem only exists in the sense that there exists a knot classification scheme based on 3-manifold classification. This is an indirect and unusable solution as it is exponential in the amount of computing time required as a function of the crossing number of the knots concerned. As many braid words represent the same braid, we are naturally lead to ask for a particularly simple representative. A very natural definition of “simple” in the case of a braid word is the least length possible; the shorter the word, the simpler it is. As length has an obvious minimum, we formulate the minimal word problem. Definition 2.31 (minimal word problem). Given a braid word α ∈ Bn , the minimal word problem asks one to find a braid word αm ∈ Bn such that αm ≈ α and L(αm ) ≤ L(α′ ) for any braid word α′ ≈ α. The word αm is the minimal word as it is, by definition, the shortest word in its equivalence class. The word problem is frequently solved by devising an algorithm which finds a unique representative for the equivalence class of the element under consideration. In many groups, this so called normal form is also of minimum length in the equivalence class so that word problem and minimal word Braids, Knots and Applications 29 problem are solved together. Examples of such groups include free groups, HNN-extensions and free products. For the braid groups, we shall find that the minimum word problem is far more complicated to solve than the word problem. In fact, under some very reasonable assumptions, the minimum word problem in the braid groups can only be solved by what amounts to a global search of the equivalence class. We shall outline these assumptions and the methods by which the global search may be done in later sections. 3 Braids and Knots 3.1 Notation for knots 1 2 4 5 Start 6 3 Fig. 14. The procedure for obtaining the Dowker-Thistlethwaite code for the trefoil. Dowker-Thistlethwaite Code When he originally invented knot theory, Tait sought to construct a table of distinct prime knots. Two tasks had to be undertaken: (1) An exhaustive list of all possible knots had to be constructed and (2) all duplicates had to be struck from that list. Tait used mainly combinatorial methods to construct an exhaustive list and then tried to eliminate duplicates by trying to deform knots into each other. Tait labelled each crossing by a letter of the alphabet. He named a knot by starting at some random point and going along it in the direction of its orientation, writing down the label of the crossings as he passed them. This string of letters is called the Tait code of a knot. Clearly a knot with n crossings has a name consisting of 2n letters. There exist a finite, though large, number of possible such names for any n. Not all possible names can arise from naming a knot, however and Tait was able to find methods to determine if a specific name was valid. Dowker and Thistlethwaite improved Tait’s methods and introduced their own naming convention [78]. Consider a knot of n crossings and start at a random point going along the knot in the order of its orientation. Name the crossings in numerical order as you pass them giving each crossing two 30 Patrick D. Bangert numerical labels (you will pass each crossing twice before returning to the staring point). Each crossing will have two numbers associated with it, i and j, say. We can construct an involution a from these by putting a(i) = j and a(j) = i. Thus we have a(a(k)) = k for any label k. Note that a reverses parity, that is if i is odd and a(i) = j, then j is even and vice-versa. We agree to write ai for a(i), S for the sequence a1 , a2 , · · · , a2n and Sodd for the sequence a1 , a3 , · · · , a2n−1 . The sequence Sodd completely determines both S and a [78]. There are 2n different possible starting points on the knot and if it has no orientation, there are two possible orientations. Not all sequences Sodd are identical. The standard sequence Sstd for a knot will be the lexicographical minimum over all possible Sodd . Consider, for example, the trefoil knot in figure 14. The starting point is labelled in the figure and we proceed to name the three crossings twice each in the order in which we encounter them. This gives rise to the involution a1 = 4, a2 = 5, a3 = 6, a4 = 1, a5 = 2, a6 = 3 (54) which results in Sstd = 4, 6, 2. Dowker and Thistlethwaite were able to determine both necessary and sufficient conditions for a sequence of number to be realizable as a knot. This algorithm is relatively simple and quick to implement and has been used by them to tabulate knots. This code is very compact and easy to obtain from a knot, but their tabulation methods focus on enumerating all possible sequences and so we ask: How may we recover the knot given the code? We note first of all that a knot of n crossings will get 2n labels (2n being necessarily even). Suppose Sstd = b1 , b2 , · · · , bn and the original involution is a. As Sstd is a standard sequence, we have that a2i−1 = bi for i ranging from one to n. Because of the definition of a, we then have a(a(2i − 1)) = a(bi ) = 2i − 1 (55) Since a reverses parity and we list only the ai for i odd in the standard sequence, all these ai are even. Thus we have recovered the entire involution a. The crossings of the knot thus get the double names i and bi for i ranging from one to n. Having gotten the complete labelling information, we can draw the crossings on our paper as double points and connect them in order of the labelling and thus retrieve the knot. We may not get the same number of crossings that we had in the original projection but this does not matter topologically. Conway’s Basic Polyhedra We constructed the Alexander polynomial in section 2.5. Starting from a knot, we must first construct a closed braid isotopic to the knot, then write down its Bureau representation and take its determinant to obtain the Alexander polynomial. In practise this process, particularly constructing a closed braid representative, is very complicated Braids, Knots and Applications • • • • • • 0 ∞ • • • • • • • • • • 1 31 −1 Fig. 15. The four elementary tangles. and time-consuming. When Conway looked for a mechanizable method, he was lead to construct a new notation for knots. From this notation, he was able to extract the Alexander polynomial so straightforwardly that he proceeded to calculate them all by hand rather than mechanize the method. Fig. 16. The universal polyhedron. A knot diagram has crossings and arcs connecting the crossings. If we were to draw small circles around the crossings and then ignore what is in the circles, we would have a template for the knot. Into the circles we could insert any of the four elementary tangles of figure 15 to generate several knots, one of which would be the original knot. The numerical names of the elementary tangles arise from their classification which will not concern us here. Conway’s notation derived from the observation that many knot diagrams are the same after the crossings have been so removed. In such a way, we may generate a large number of different knots starting from one such knot template and inserting different tangles into different slots. Conway called these templates basic polyhedra and was able to show that eight basic polyhedra are enough to generate all the different knots up to and including 11 crossings [61]. The number of different basic polyhedra needed to generate the knots of higher crossing number n increases sharply with n and as the polyhedra lack pattern, the next one may not easily be generated from the previous ones. This problem gave rise to the idea of the universal polyhedron to be discussed next. The Universal Polyhedron The universal polyhedron P (i, j) is defined by figure 16. It has i rows and j columns of vertices which will be filled with elementary tangles and which are connected by edges. It can be shown that 32 Patrick D. Bangert any knot can be represented by some P (i, j) and that we may write this down in a matrix form,   p11 p12 · · · p1j  p21 p22 · · · p2j    (56) P (i, j) =  . . . .   .. .. . . ..  pi1 pi2 · · · pij with the elements pkl ∈ {1, −1, 0, ∞}. For example, the knots up to and including six crossings (see figure 8) are given by   −1 −1 31 = 01 = (∞) −1 ∞  
 1 1 1 41 = 51 = 1 1 1 1 1 0 −1 ∞     ∞ 0 −1 −1 ∞ 0 1 (57) 52 =  −1 −1 −1  61 =  1 1 0 ∞   0 1 ∞   0 −1 −1 ∞ 1 1 1 ∞ 1 1 1 62 =  0 −1 −1 ∞  63 =  0 −1 ∞  0 1 ∞ 0 1 1 0 Theorem 3.1. Every regular projection of any knot may be represented by the universal polyhedron P (i, j) for some i and j all the vertices of which contain elementary tangles. Proof. A regular projection of a knot is characterized by a finite number n of double points and 2n arcs which connect the double points in a specific manner. For sufficiently large i and j, the polyhedron P (i, j) can accommodate all double points in the form of ±1 tangles and can achieve the desired connection of these by placement of 0 and ∞ tangles into it. This is obvious because the 0 and ∞ tangles represent horizontal and vertical connectors in the polyhedron. Because this connection may be achieved without ±1 tangles, it is clear that no further components, with the possible exception of unknots, are created. Thus what remains to be shown is that no unwanted unknots will be created. There are ij vertices and 2ij edges connecting them in the empty polyhedron P (i, j). Eliminating one vertex by a 0 or ∞ tangle, eliminates two edges. Apart from the ±1 tangles of which there are n, the final polyhedron will contain ij − n tangles of type 0 and ∞ which will have eliminated 2(ij − n) edges from the original polyhedron, leaving exactly 2n edges which are needed to connect the double points. Thus there is no extra edge left over which could possibly form an extra component. Therefore any knot may be represented using the basic polyhedron P (i, j) and elementary tangles. Braids, Knots and Applications 33 3.2 Braids to Knots Suppose we have a braid b given by a braid word and we want to denote the knot that is isotopic to its closure in the standard notations. The universal polyhedron (figure 16) turned through π radians becomes a closed braid template if every vertex is filled with tangles of the types 1, -1 and 0. The n-braid b will be specified by a function b(t) = σi±1 which gives the tth Artin generator of b for 1 ≤ t ≤ c where c is the number of crossings in the braid. The map ξi will map an Artin generator σj±1 to the elementary tangles 1 and -1 if the exponent of the Artin generator is 1 and -1 respectively and i = j and will map any Artin generator to the elementary tangle 0 otherwise, ξi (σi ) = 1
 ξi σi−1 = −1
 ξi σj±1 = 0 for i = j (58) (59) (60) The closed braid b can be contained in the polyhedron P (n − 1, c) with pij = ξi (b(j)) with 1 ≤ i < n and 1 ≤ j ≤ c. For example, b = σ13 . Then b(i) = σ1 for i = 1, 2, 3 and we get
 σ13 = 1 1 1 (61) which correctly represents the trefoil knot. In the next section we will generalise this example to an infinite family of knots known as torus knots. 3.3 Example: The Torus Knots The torus knots are an infinity family of prime knots which have particularly simple properties and are frequently used as examples in knot theory texts. The connection between braids and knots is readily illustrated in the case of torus knots and this is what we shall do below. Definition 3.2. Given two co-prime (no common factors apart from unity) integers p and q, the torus knot Tp,q is constructed by wrapping a closed curve around the surface of a torus such that it encircles it p times meridionally and q times longitudinally (respectively the short and the long way around). Torus knots are completely characterized by the two integers p and q. They are invertible (isotopic under switching the orientation) and chiral (not isotopic to their mirror image) [48]. The fundamental group of their complements is given by π1 (Tp,q ) = {a, b} : ap = bq  [118] from which we may recognize the requirement that p and q be co-prime. It may be shown that Tp,q = Tq,p . The torus knots are among the few knots for which the minimal number of crossing-switches required to transform the knot into the unknot, i.e. the unknotting number, is known; it is (p − 1)(q − 1)/2 [1]. They are also 34 Patrick D. Bangert among the few knots for which the minimal number of crossings in any projection, i.e. the minimal crossing number is known; it is min[p(q −1), q(p−1)] [211]. Exercise 3.3. Given the above fundamental group show that p and q must be co-prime to generate a distinct knot and if co-prime completely classify the torus knots. Exercise 3.4. Show that Tp,q = Tq,p . The simplest example of a torus knot is the trefoil. It is not the unknot even though Tp,1 for any p would be isotopic to the unknot but p and unity are not coprime. The trefoil knot is T3,2 and the knot 51 is T5,2 . In general, q the closed p-braid (σp−1 σp−2 · · · σ1 ) is isotopic to the torus knot Tp,q . This is easily seen by picturing Tp,q on an actual torus. We cut the torus across a random meridian (the short way around) and straighten it into a cylinder. The remainder will be the braid above. Moreover, there exists no closed braid representative of Tp,q with less than p strings so that p is the braid index of Tp,q . Exercise 3.5. Draw T3,4 on a torus and then cut the torus across a meridian 4 and show that T3,4 is isotopic to the closed braid (σ2 σ1 ) . Exercise 3.6. Show that the Alexander polynomial of Tp,q is given by △Tp,q = (1 − tpq )(1 − t) (1 − tp )(1 − tq ) (62) As we have seen, any torus knot can be represented as the p-braid b = q (σp−1 σp−2 · · · σ1 ) . By using the method of the last section, we can represent Tp,q in the polyhedron P (p − 1, q(p − 1)).   1 1 ··· 1  ··· ··· ··· ···      1 1 ··· 1 (63) Tp,q =     1 1 ··· 1 1 1 ··· 1 The Dowker code may be obtained from a knot represented in our notation by simply walking through the polyhedron and labelling the crossings. As the polyhedron is structured, this walk is perfectly definite and can be programmed easily on a computer. 3.4 Knots to Braids I: The Vogel Method A braid is more structured than a knot and so the transition from knot to closed braid is harder to effect than the reverse. There exists a simple method Braids, Knots and Applications or 35 → Fig. 17. Both types of crossing have to be reconnected in the shown way in order to obtain a diagram of Seifert circles from a knot diagram. due to Vogel [198] which we shall present without proof in this section. Suppose we are faced with a knot diagram D which we want to convert into a closed braid. For this method, we shall have to view D in a variety of ways. From the diagram, we can get to the projection P of the knot onto the plane by viewing each crossing as a double point and thus ignoring over and undercrossing information. From D, we can construct a diagram S by reconnecting each crossing in D in the manner shown in diagram 17. The diagram S will contain a number of unknots which we will call Seifert circles. Using these constructions, we can define the crucial concept in Vogel’s method. Definition 3.7 (admissible triple). Let f be a face of P and a and b be two edges of P . The triple (f, a, b) is called an admissible triple if and only if it satisfies: (i) a and b are contained in different Seifert circles and (ii) a and b have the same orientation with respect to any orientation of ∂f , the boundary of f . It is shown in [198] that the following algorithm will transform any knot diagram D into a diagram of a closed braid. Algorithm 3.8 Input: A knot diagram D. Output: A knot diagram D′ ambient isotopic to D and in the form of a closed braid. Fig. 18. The two types of admissible triples are shown in the solid curves and the form into which they should be transformed is shown in the dashed curves. 36 Patrick D. Bangert 1. Determine if D has an admissible triple. If yes, continue. If no, D is in the form of a closed braid and the algorithm is done. 2. Admissible triples can come in the two flavours shown in the solid curves in figure 18. Each admissible triple detected, is to be transformed (via a Reidemeister move type 1) from the solid curve to the dashed curve in figure 18. Such a transformation will be called an elementary transformation. Then go back to step 1 of the algorithm. It can be shown that algorithm 3.8 always terminates after at most (s − 1)(s − 2)/2 elementary transformations where s is the number of Seifert circles in S. The braid which this algorithm generates has at most n + (s − 1)(s − 2) crossings where n is the number of crossings in D as the elementary transformation adds two crossings each time it is applied. It is unclear whether algorithm 3.8 is confluent, that is whether the order in which we perform elementary transformations should two (possibly overlapping) triples be simultaneously admissible changes the final outcome. 3.5 Knots to Braids II: An Axis for the Universal Polyhedron Having constructed a new notation for knots, we wish to solve the problem of how to extract a closed braid from the matrix which is isotopic to the knot described by the matrix. A few algorithms have been constructed in the past, which convert a knot into a closed braid but they are difficult to implement because they depend upon topological deformation of the knot projection [129] [31]. The best known algorithms have been implemented [198] [219] and have complexity O(n2 ). We shall present an algorithm which achieves the conversion with complexity O(n), increases the number of crossings only in a few cases (and then only by a few crossings) and uses a linearly bounded number of strings. There exists no algorithm to calculate the number of strings which are at least necessary to describe a specific knot — the braid index of the knot. Because of this, it is not possible to say how close to the minimum the number of strings used by our algorithm is. The number of crossings is sometimes increased because it has been found that there are knots for which any closed braid representative has more crossings than the minimal knot diagram; the knot 5.1 in the standard tables is the simplest example of this [174]. Our algorithm is valid both for oriented and unoriented knots. An Example Alexander’s theoremiAlexander’s theorem was proven by showing that every knot can be deformed into a form where the knot loops around an axis a finite number of times without local maxima or minima with respect to that axis. If we cut the string along the axis in one place, we obtain a braid. The gluing back of the cut constitutes the canonical closure. Thus as far as the canonical closure is concerned, the finding of an appropriate axis is the key. Having obtained a canonically closed braid which is equivalent to Braids, Knots and Applications 37 a knot, we may obtain a plait from it by considering the closure curves part of the braid diagram and moving them into the middle of the braid diagram. The next section gives an example of this. A Fig. 19. The trefoil knot with an axis for braiding it. For the rest of this section, we are going to work through an example of our method. Consider the trefoil knot in figure 19. We have drawn an axis through it by the following method: (1) We drew a line through the projection of the trefoil which intersects every region of the plane at least once, (2) begins and ends in the infinite region and then (3) assigned the under and overpasses of the knot under and over the axis by traversing the knot from a random starting point (point A in the figure) while (4) assigning the passes alternately as we met the crossings of axis and knot. Next we perform a coordinate transformation from the knot reference frame (figure 19) to the axis reference frame in figure 20 by pulling the axis straight. We can easily observe from figure 20 that the axis is valid; i.e. if we traverse the knot starting at A we will travel around the axis without local maxima or minima permanently in a clockwise direction. If we now cut the knot at those points at which it overcrosses the axis and lay out the ends carefully to either side, we shall obtain the braid σ1−1 σ2−1 σ1−1 σ2−1 shown in figure 21 (a). To get back to the trefoil from this, we perform the canonical closure which is identical to sealing the cuts made above. This is shown in figure 21 (b). This knot has four crossings and is ambient isotopic to the trefoil thus there is some inefficiency in our braid representation (note however that there exist knots for which the most efficient braid representation contains more crossings than their most efficient knot projection [174]). We note that we may lift the arc labeled in figure 21 (b) to remove one crossing. This move also removes a string and so we obtain the braid of figure 21 (c). This braid has two strings and three crossings, it is thus the most efficient representation of the trefoil as the trefoil must have at least this many strings and crossings. 38 Patrick D. Bangert A Fig. 20. The trefoil knot as it appears after the axis has been straightened from figure 19. For reference the point A has been labeled here again. We conclude that the closure of the braid σ1−1 σ1−1 σ1−1 is ambient isotopic to the trefoil knot. Note that we may turn the entire figure 21 (c) about a vertical axis through its center and thus obtain the result that the braid σ1 σ1 σ1 is ambient isotopic to the trefoil also; this, finally, is the well-known braid representation of the trefoil knot. This is the prototype for a general method which we shall develop below. Platting a Knotiplait The diagram of a knot which is expressed as a closed braid may be naturally divided into two parts: the braid and the closure. The most important feature of the braid part, for our purpose, is the requirement that all strings be monotonic increasing in the vertical coordinate, that is they may only go side to side and never double back on themselves. In this light, consider turning the polyhedron P (i, j) clockwise by π/2. If the polyhedronipolyhedron does not contain any ∞ tangles, this is already a canonically closed braid. However, in general, the polyhedron will contain ∞ tangles. Note that the rotation will make the ∞ tangles look like 0 tangles. In an effort to rid ourselves of the ∞ tangles, we take the top string in the ∞ tangle and move it all the way to the bottom of the knot diagram and move the bottom string all the way to the top. In this way, we have created two extra strings in the braid which are closed in the plait manner. If we do this for all ∞ tangles, we will have a valid braid in the center of the diagram but the closure mechanism will be a hybrid between the canonical and plait methods. In order to rectify the situation, we move the strings which are closed in a canonical manner into the center of the braid diagram, thereby Braids, Knots and Applications 39 Lift (a) (b) (c) Fig. 21. The braid which is extracted from figure 20 by cutting the trefoil knot at its overcrossings over the axis and laying out the ends is displayed in part (a). The closure of this braid is part (b). If we lift the arc labeled in part (b) we obtain the braid in part (c). See discussion in the text. creating more strings and more crossings. Once this has been done, we have a fully valid braid closed in the plait manner which is ambient isotopic to the knot we started with. Figure 22 shows the process of converting the unknot   −1 1 (64) U= ∞ −1 into the braid σ2 σ4−1 σ3 σ4 σ5−1 σ6−1 σ4−1 σ5−1 σ4 σ6 closed in the plait manner. This procedure is valid generally and clearly represents a readily implementable algorithm for transforming a knot given in our notation into a plait. If the original knot is given in the polyhedron P (i, j) and has k tangles of type ∞, then the number of strings required in the plait is 2(i + k + 1) but the number of crossings depends upon the exact configuration. Laying the Axis As mentioned before, the transformation of a knot projection into a canonically closed braid centers around finding an appropriate axis for the string to wind around. This was the central point of Alexander’s theoremiAlexander’s theorem which proves that such an axis may always be found. A ready method for finding an axis is given in the following algorithm. Algorithm 3.9 Input: A knot projection. Output: A knot projection with an axis around which the knot winds without local maxima or minima. 1. Begin with enumerating the regions into which the knot projection divides the plane, suppose there are R of these. 2. Choose two arbitrary points in the infinite region and call them A and B. 40 Patrick D. Bangert Fig. 22. The conversion of a knot into a plait. 3. Draw a line L connecting A and B in such a way that the line intersects every region at least once. 4. Choose a random point on each of the knot’s components and traverse the knot in the direction of the orientation once for each component starting at the chosen point. While traversing label each intersection of L with the knot alternatingly with a + or − sign starting with +. 5. Interpret each + crossing as an overcrossing of L over the knot and each − crossing as an undercrossing of L under the knot. The line L oriented from A to B is then a valid axis. This algorithm may clearly be applied to our polyhedron P (i, j). However we have the problem of the regions which depends upon the exact configuration of the knot. This can be solved by forcing the line L to intersect every region in the polyhedron and therefore intersecting some regions of the knot more than once. This is unfortunate but unavoidable if we are seeking a general solution of the problem. The manner in which this may be done most economically is illustrated in figure 23. The line L is the dotted line beginning at point A and finishing at point B. If the polyhedron has an odd number Braids, Knots and Applications 41 C A B Fig. 23. The axis of the braid through the polyhedron P (i, j)ipolyhedron. of columns (as the one in figure 23), then the line L is best described by the dotted line in figure 23. If however, the polyhedron has an even number of columns, then the line L is best described by the dotted line in figure 23 from point A to point C and then the dashed line from point C to point B. If algorithm 3.9 is correct then a line drawn in a general polyhedron P (i, j) according to this example is a valid braiding axis. We may find an axis which passes through every region exactly once, if possible, by the following algorithm. Algorithm 3.10 Input: A matrix describing a knot in our notation. Output: A matrix describing the regions of the knot. Each element of the matrix receives a label from 1 to R, the number of regions. This gives complete information about which regions of the polyhedron are connected and how many there are. 1. Begin at the top left of vertex (1, 1) and follow the boundary downwards, as for counting regions, the orientation of the knot does not matter. Mark the region (0, 1) with a 1, the current marker, in the region matrix. 2. In following the boundary, one will come to vertex (1, 1); we assess its value and continue. If we stay in the same region of the polyhedron we continue, if we enter a new region of the polyhedron, then this new region of the polyhedron belongs to the same region of the knot as the previous one and thus we mark it with the current marker in the region matrix. The whole issue at hand is that the regions of the polyhedron are known while we wish to gain knowledge of the regions of the knot. 3. We continue to follow the boundary until we reach the point of origin. 4. We search the matrix for an unmarked region. If there exist unmarked regions, we increment our current marker and choose one of the regions 42 Patrick D. Bangert as our new starting region and choose a point upon its boundary as our new starting point. Then, we repeat the algorithm from step 1, marking the region with the current marker. 5. Once no unmarked region of the polyhedron exists, the algorithm is finished. The largest marker used in the matrix which we have obtained is clearly the number of regions of the knot. Furthermore, since all connected regions are labeled with the same marker, we have a complete knowledge of which regions of the polyhedron belong to the same region of the knot. Algorithm 3.11 Input: A knot projection given in our notation. Output: An axis which passes through every region exactly once, if this is possible. If not the output is an axis which passes through each region at least once. 1. Get the region information as prescribed in algorithm 3.10. 2. Construct a graph in which each region is symbolized by a node and two nodes are connected by an unweighed edge if they are adjacent in the plane. 3. A Hamiltonian circuit is then a path which passes through each region, that is node, exactly once starting in the infinite region and returning there. If a Hamiltonian circuit exists, so does an optimal axis. If no Hamiltonian circuit exists, we find an axis using algorithm 3.9 which gives an axis which passes through every region at least once. The advantage is that we will generate a braid with less strings but the Hamiltonian circuit problem is NP-completeiNP-complete and so the execution of algorithm 3.11 is exponential (unless we use an approximation algorithm or it is shown that P = NP). This fact lends further weight towards the usefulness of algorithm 3.9. The primary usefulness of this algorithm originates in the fact that the laying of the axis does not depend upon the exact knot configuration, only the labelling does. Before we continue, we prove that algorithm 3.9 always yields a valid axis, this essentially amounts to proving Alexander’s theoremiAlexander’s theorem. Theorem 3.12. Given any knot projection, algorithm 3.9 will find an axis about which the knot is without local maxima or minima. Proof. Alexander’s theorem [11]iAlexander’s theorem states given a knot projection, it is possible to deform it with respect to a point P in the projection plane that after the deformation a point A which travels along the knot in the direction of its orientation will travel around the axis defined by P (the axis is a line perpendicular to the projection plane intersecting it at P ) in a constant fashion, either clockwise or counterclockwise, for the entire circumnavigation of the knot. We wish to do the opposite, namely to deform the axis around the knot projection to achieve the same ends. We can imagine the process of laying the axis as akin to sewing in which we move the needle Braids, Knots and Applications 43 up from and down onto the plane. Morton [157] has constructed a similar method to ours which he calls “threading.” The knot divides the plane into several regions. If the axis does not intersect a particular region, the point A will change course during traversing the knot and so the axis must intersect each region. It is however clearly only necessary for the axis to intersect the region once. Choose a line in the plane which intersects the axis. With respect to this line we can define an angular coordinate θ going around the axis. As point A must travel around the axis in a constant fashion it must, after it passes θ = 0, reach θ = π before it once again reaches θ = 0. This shows that the axis, in the projection plane, must over and undercross the knot alternately with respect to A. This fulfills the requirements of an axis and these are assured by algorithm 3.9 and thus the theorem is proven. ✷ Getting the Braid Having obtained the axis, we must now simply put together all the pieces and construct the braid. This will be done via the following algorithm. Algorithm 3.13 Input: An axis L in a knot projection given in P (i, j) using our notation. Output: A braid the canonical closure of which is ambient isotopic to the given knot. 1. Consider an empty polyhedron P (i, j)ipolyhedron and label each edge by the row and column index of the vertex out of which it is emerging on the right side giving it the further label a if it is the top edge and b if it is the bottom edge. That is the top right hand edge coming out of the vertex (1, 1) would be (1, 1)a . 2. All edges which intersect the axis L at a positive crossing are to be numbered in order starting at point A; suppose there are k of these. 3. Starting at the numbered edges, use the traversal algorithm to follow each edge around the knot until another positive crossing with the axis L. All edges encountered are to be labelled with the same number as the original edge. 4. When all edges are numbered, we have identified the individual strings of the braid and numbered them in order. Assign a distance value of 1 to each edge in the polyhedron. 5. Traverse the knot again as in step 3 but this time stopping at each double point and extracting which labelled string passes over which other labelled string and at which distance value this occurs. 6. When the whole has been traversed, we have a list of crossings specifying which strings are involved, which string crosses over the other and at what distance from the bottom of the braid the crossing occurs. This information may be used readily to construct a colored braid, which may be converted easily into an Artin braid word. 44 Patrick D. Bangert 7. We assess the string labels around the knot and calculate the permutation associated with the braid which winds around our axis. If this permutation is different from the permutation of the braid which we obtained in step 6, the residual permutation must be added to this braid in the form of extra crossings. The number of crossings is increased in some circumstances by a small amount in step 7 of the algorithm. It is a fact that there exist knots of minimal crossing number n which have closed braid representatives all of which have crossing numbers greater than n [174]. Hence, step 7 is not a deficiency of the algorithm 3.13 but a fundamental necessity. It is clear from Alexander’s theoremiAlexander’s theorem[11] that this algorithm works. The number of strings used is the number of positive crossings of the axis with the knot which is equal to half the number of crossings. The number of crossings of the axis with the knot is " j−2 # ! j odd 4i + (2i + 2) 2
(65) Nc = 2i + (2i + 2) j−2 + 2 j even 2 where ⌊x⌋ is the greatest integer less than x. An analysis of the possibilities in oddness and evenness of i and j reveals that Nc is always even which is good since we must have an equal number of positive and negative crossings. Algorithm 3.13 therefore finds a braid with a number of strings which scales linearly in the number of rows and columns necessary to represent the knot. It is conceivable that a more economical way of laying an axis may be found using algorithm 3.11 but this has an exponential complexity. The number of strings may be reduced after the braid has been found using Markov’s theoremiMarkov’s theorem. The determination of the regions, the laying of the axis, the labelling of the axis crossings, the labelling of the edges and the extraction of the double point information all take a time proportional to the number of vertices in the polyhedron ij. The building of the braid from the crossing information takes time proportional to ij. Therefore the entire algorithm to proceed from a knotiknot projection to a canonically closed braid has complexity O(ij). This algorithm succeeds in being readily implementable and in constructing a braid which is reasonably small. 3.6 Peripheral Group Systems of Closed Braids In this section we will investigate the peripheral group system of the closure of the fundamental braids. The peripheral group system is a complete invariant of knots and figures largely in knot theory. The fundamental braid words are very important in braid theory and we choose them for this investigation for that reason and that the closure of ∆3 is the Hopf Link and the closures of the other fundamental braids look very similar to Hopf Links. In fact so similar Braids, Knots and Applications 45 that we can regard the class of knots defined by the closure of fundamental braids as a generalization of the Hopf Link. Another generalization of the Hopf link has been investigated in the literature [52]. We shall see that our methods developed here can be extended beyond fundamental braid words to all braid words. The Fundamental Group Consider a space X and a point x0 ∈ X which we shall call the base point. In the space X, we may construct loops, i.e. paths from x0 to itself. Definition 3.14 (fundamental group). The group that consists of the loops at x0 in the space X with respect to the homotopy equivalence relation (continuous maps from one set of loops to another) is called the fundamental group of the space X and is denoted by π1 (X, x0 ). It can be shown that if X is path connected, the choice of base point does not matter. All spaces we are about to consider are path connected and so we shall drop the specification of the base point and denote a fundamental group by π1 (X). Definition 3.15 (knot complement). The complement of a knot K with respect to a space X is the space X − V (K) where V (K) represents a tubular neighborhood of the knot K. Definition 3.16 (knot group). The group of a knot K, denoted by π(K) is the fundamental group of the complement of the knot with respect to the space X = R3 (sometimes X is taken as S 3 but it can be shown that the two fundamental groups arising from X = R3 and X = S 3 are isomorphic). Fig. 24. The labelling of the trefoil knot in order to yield a presentation for the fundamental group of its complement. Consider a knot K and its complement R3 − V (K). In this space, choose a point x0 as the base point (the choice is arbitrary as the space is path connected). Consider the projecting cylinder Z ∈ R3 which contains all of 46 Patrick D. Bangert V (K) and is constructed such that the projection of K onto the plane z = 0 in R3 contains at most double points. This projecting cylinder will contain n self-intersections if K has n crossings. Label these self-intersections by ai and the sections of Z between the ai by si . Z is to be oriented in order to match the orientation of K. Denote a loop from the base point to itself which goes exactly once around si and no other sj by pi . From this construction, it is clear that all loops (with respect to homotopy equivalence) can be constructed as products of the loops pi . This can easily be seen by the fact that any path in the complement can be continuously deformed into a product of loops pi . Thus π(K) is generated by the loops pi . For an example see figure 24. Having gotten the generators, we need the relations to obtain a group presentation. Consider the loops encircling the ai and join them to the base is contractible (homotopic to point by a path ci , then it is clear that ci ai c−1 i the trivial loop). The word in the group corresponding to this loop is then a relation in the group. The presentation so obtained is called the Wirtinger presentation of the knot group. This discussion proves that the following algorithm to obtain it is correct. • xl 7• ' xi x ' k • • (1) • xl ' w • • xi x ' k • (2) Fig. 25. The two possible forms of double points in the diagram of an oriented knot. The crossing of type 1 has characteristic epsilon = 1 and the type 2 has characteristic ǫ = −1. Algorithm 3.17 (Wirtinger presentation of the knot group) Input: A knot projection of a knot K. Output: The Wirtinger presentation of π(K). 1. Label all overcrossing arcs in the projection by gi for 1 ≤ i ≤ n where n is the number of double points in the projection starting at any point and then assigning the labels in order corresponding to the orientation of the knot. 2. For each double point, determine its characteristic sign, see figure 25. The characteristic can be most easily determined by the “right hand rule” which says that if you spread your thumb at right angles from your fingers and point it along the overcrossing arc, and if your finger point along the undercrossing arc, the characteristic is 1 and -1 otherwise (along being taken to mean along the orientation of the knot). 3. π(K) = {g1 , g2 , · · · , gn } | {r1 , r2 , · · · , rn } where the relators are given by ǫ −ǫ rj = gj gi j gk−1 gi j and ǫj is the characteristic of the crossing associated with rj . Braids, Knots and Applications 47 It can be seen that one relation can be derived from the others and thus a knot group has deficiency one. The knot group is an invariant of knots as is trivially seen by definition but it is not complete. Thus if two knots have isomorphic knot groups they are not necessarily isotopic. However if they have non-isomorphic groups, the knots are distinct. The unknot U has the knot group constructed by a single generator and no relations, i.e. π(U ) = Z, the infinite cyclic group. It is a practical observation that the Wirtinger presentation can often be simplified considerably in that some generators are removable [97]. In particular, π(K) for the torus knot Tp,q , which is a knot that winds around a torus p times the short way around (meridionally) and q times the long way around (longitudinally), is given by π(K) = {a, b} : ap = bq  [118]. (a) (b) Fig. 26. The construction of the Wirtinger presentation of the fundamental group of the complement of the (a) square and (b) granny knots. The Square and Granny Knots The knot group is not a complete invariant. We wish to illustrate this by an example. The knot groups of the square S and granny G knots are isomorphic but the knots are distinct. The demonstration of this fact will occupy this section. For a picture of these knots, see figure 26. We follow algorithm 3.17 to compute the knot groups. Both knots receive six generators and six relations. All crossings in the granny knot have characteristic -1 as well as three crossings in the square knot, the other crossings in the square knot have characteristic 1. We thus write down the Wirtinger presentations. 48 Patrick D. Bangert  {g1 , g2 , g3 , g4 , g5 , g6 } | g6 g2 = g2 g1 ; g5 g1 = g1 g6 ; π(G) = g1 g6 = g6 g2 ; g2 g4 = g4 g3 ; g4 g3 = g3 g5 ; g3 g5 = g5 g4   {g1 , g2 , g3 , g4 , g5 , g6 } | g6 g2 = g2 g1 ; g5 g1 = g1 g6 ; π(S) = g1 g6 = g6 g2 ; g3 g2 = g2 g4 ; g2 g4 = g4 g3 ; g4 g3 = g3 g5  (66) (67) Three generators may be defined in terms of the other three in both groups and after some simple manipulation we obtain   {g1 , g2 , g3 } | π(G) ≈ π(S) = (68) g1 g2 g1 = g2 g1 g2 ; g2 g3 g2 = g3 g2 g3 (a) (b) Fig. 27. The (a) square and (b) granny knots from figure 26 transformed into closed braids. Even though the knot groups are isomorphic, the knots are distinct. This can seen by deforming the knots into closed braids and computing their Jones polynomials. The transformation is straightforward and the result is shown in figure 27. Reading off from the figure, we obtain that G ≈ σ1−3 σ23 ; S ≈ σ13 σ23 (69) Recalling the algorithm to compute the Jones polynomial yields 

 2

VS (t) = t + t3 − t4 VG t = 3 − t3 + t−3 + t2 + t−2 − t + t−1 ; (70) and we see that the knots are distinct proving that the fundamental group is a incomplete invariant. Braids, Knots and Applications 49 Peripheral Group System The fundamental group is not a complete invariant but it is possible to refine our methods to construct a complete invariant from the fundamental group, the peripheral group system. This is the only complete invariant of knots that we can readily compute for all knots. The problem is that distinguishing knots via the peripheral group system requires distinguishing groups with respect to isomorphism which is known to be an undecidable problem [2, 3, 172]. The complement of a knot is uniquely specified (up to isomorphism) by its peripheral group system which consists of the fundamental group and a few subgroups thereof (this is Waldhausen’s theorem [201], see [104] for a more accessible proof). It is however known that the word problem for any fundamental group of any knot is solvable [202]. If the knot is alternating, the conjugacy problem is also solvable [12]. We define the linking number of two curves a and b, denoted by lk(a, b) as the weighted sum of the characteristics ǫ of each crossing. We define a meridian mi and a longitude li of a knot component Ki by requiring the following properties: (1) mi and li are oriented, polygonal, simple and closed curves in ∂V (Ki ), the boundary of the thickened neighborhood of Ki which we denote by V (Ki ), (2) mi and li intersect in exactly one point, (3) mi is null homologous (mi ∼ 0) in V (Ki ) and li ∼ Ki in V (Ki ), (4) li ∼ 0 in C (Ki ) and (5) lk (mi , Ki ) = 1 and lk (li , Ki ) = 0 in S 3 . The above five properties define mi and li uniquely up to isotopy on the boundary of V (K) [48] (see figure 28 for an illustration). The meridian-longitude system pair M(K) for a j-component knot K is the pair of sets ({m1 , m2 , · · · , mj }, {l1 , l2 , · · · , lj }). Interestingly, if the longitude is trivial (equivalent to the identity element) in the knot group, the group is infinite cyclic; i.e. the group is isomorphic to the fundamental group of the complement of the unknot. Furthermore, the meridians and longitudes commute with each other in any knot group [118]. Fig. 28. The thick curve displays the trefoil knot with an orientation. The thin curve which is parallel to the trefoil knot is the longitude; the orientation of the longitude is the same as the knot. The thin curve encircling both trefoil and longitude at the top left hand corner is the meridian. Note that the five conditions given in the text are fulfilled by these curves. 50 Patrick D. Bangert The meridians can be taken as the Wirtinger generators of the knot group. The longitude of a knot may be read off the projection very easily. Begin with the first generator and traverse the knot in the direction of its orientation. Whenever undercrossing an arc, write down the generator of the undercrossed arc to the power of the negative of the characteristic of the crossing. After the full traversal, append as many copies of the initial generator (or its inverse) to make the total sum of exponents equal to zero. The meridians and longitudes of M(K) may be considered to be elements of π(K) by choosing a path pi in C(K) from the base point x0 to the (unique by definition) point mi ∩li for each i. Then the subgroup mi , li  of π(K), generated by mi and li is independent of the choice of pi up to conjugation. The peripheral group system of a j-component knot K is p(K) = (π(K); M(K)). By an isomorphism φ between two peripheral group systems p (K) ≈φ p (K ′ ), we mean π (K) ≈ π (K ′ ) such that φ (mi ) = m′ i and φ (li ) = l′ i for all i. It can be shown that for any two knots K1 and K2 , p (K1 ) ≈ p (K2 ) if and only if K1 ≈ K2 [118]. If we restrict attention to prime knots of a single component, we have π (K1 ) ≈ π (K2 ) if and only if K1 ≈ K2 [118]. Thus the problem of knot isotopy can be transformed into the problem of peripheral group system isomorphism. Since it is not possible to determine, in general, if two groups are isomorphic, this does not solve the knot classification problem. 4 Classification of Braids and Knots In this section we will discuss the word, conjugacy, Markov and minimal word problems introduced in section 2.8 in detail. There are a number of distinct solutions to the word and conjugacy problems in Bn each of which has special features and gives additional insight into the problem and braids in general. We shall discuss briefly the solutions due to Garside as they are historically the most important ones and have had ground-breaking influence on the field. Then we shall discuss a new solution to both problems. The Markov problem is unsolved at present but we will discuss some of its features and why it is so difficult. Lastly, we will present a number of results about the minimal word problem. It is computationally expensive to solve (NP-Complete) and so we present a heuristic algorithm and some simulation approaches and interesting results arising from them. 4.1 The Word Problem I: Garside’s Solution In the Artin presentation, the word problem asks whether two n-braid words η α = σaǫ11 σaǫ22 · · · σaǫoo and β = σbη11 σbη22 · · · σbpp such that 1 ≤ ai , bj < n and ǫi , ηj = ±1 for 1 ≤ i ≤ o and 1 ≤ j ≤ p are equivalent to each other (denoted α ≈ β) under the equations σi σj ≈ σj σi for |i − j| > 1 and σi σi+1 σi ≈ σi+1 σi σi + 1. This problem is traditionally approached by trying to find an Braids, Knots and Applications 51 algorithm which will use the two braid group relations to construct, from any braid, a unique normal form. A unique normal form α of the braid α and the corresponding form β of the braid β are exactly equal (α = β) if and only if α ≈ β. Artin was the first to describe a normal form to solve the word problem for braids [17] but the normal form has a exponentially growing number of crossings in terms of the original number of crossings and so is not as useful as other methods. Garside constructed another unique normal form which has many important properties [94]. Garside’s method begins with the crucial proposition, Proposition 4.1. For any σi−1 , we have σi−1 ≈ ∆−1 n ∆n−1 dn−1,i+1 di−1,1 . Proof. Since ∆n = a1,n−1 a1,n−2 · · · a1,2 a1,1 and R (∆n ) ≈ ∆n , we have −1 −1 −1 −1 −1 ∆−1 n ≈ dn−1,1 dn−2,1 · · · d1,1 . Thus dn−1,1 ≈ ∆n d1,1 d2,1 · · · dn−2,1 . From the definition of di,j , we have σi−1 ≈ σi−1 σi−2 · · · σ1 d−1 n−1,1 σn−1 σn−2 · · · σi+1 ≈ ≈ ≈ ≈ ≈ ≈ di−1,1 d−1 n−1,1 dn−1,i+1 di−1,1 ∆−1 n d1,1 d2,1 · · · dn−2,1 dn−1,i+1 −1 ∆n an−i+1,n−1 d1,1 d2,1 · · · dn−2,1 dn−1,i+1 ∆−1 n an−i+1,n−1 ∆n−1 dn−1,i+1 −1 ∆n ∆n−1 di−1,1 dn−1,i+1 ∆−1 n ∆n−1 dn−1,i+1 di−1,1 for any i, which proves the proposition. (71) (72) (73) (74) (75) (76) (77) ⊓ ⊔ Together with the fact that σi ∆n ≈ ∆n σn−i (see proposition 2.23), this ′ means that we may put any n-braid word α into the form α ≈ ∆−q n α where ′ α is a positive braid word and q the number of inverse generators present in α. Next we need to construct the diagram D(α′ ) of the positive braid α′ . Definition 4.2. The Cayley diagram D(b) (or diagram for short) of a positive braid word b is the set of all braid words equivalent to b. Because of proposition 2.25 which states that two positive and equal braid words are positively equal, the following algorithm obviously constructs D(b) given a positive b. Algorithm 4.3 Input: A positive braid word b. Output: The diagram D(b) of b. 1. We define the set D0 (b) by D0 (b) = {b}. 2. The set Di (b) is obtained from the set Di−1 (b) by adding to Di (b) all the braid words which can be obtained from any member of Di−1 (b) by applying any braid group relation exactly once and the deleting all those which are already contained in the sets Dj (b) for 0 ≤ j < j. 52 Patrick D. Bangert 3. Step 2 is recursively applied until an integer m is reached for which Dm (b) = ∅. It is obvious from the construction of the sets that a finite m always exists. 4. The set D(b) is then the union of the Di for 0 ≤ i ≤ m. We note that, in general, D(b) contains a number of elements which grows exponentially both with the length and number of strings in b. Once D(α) has been fully enumerated we select from it a word of the form ∆pn α′′ for which p is maximal. Then we construct D(α′′ ) and choose the braid word α′′′ from D(α′′ ) which has the lowest integer associated with it, the decimal expansion of which is given by the concatenation of the generator indices in the braid word (σ2 σ1 σ3 has associated integer 213). The braid word α′′′ is called the base of the diagram D(α′′ ). We define the Garside normal form of the braid α to be αG = ∆np−q α′′′ . We refer to p − q as the Garside exponent and to α′′′ as the Garside remainder. It can be shown that the Garside normal form is a unique normal form and that thus two n-braid words α and β satisfy α ≈ β if and only if αG = βG . Executing this method in practise is time-consuming due to the size of the diagram. There exists an efficient polynomial-time algorithm to extract the maximal number of ∆n from α′ due to Jacquemard [108] which we shall not present here. Exercise 4.4. Find the Garside normal form for the 3-braid α = σ2 σ1−1 σ2 σ1 σ2 (the solution is below). ′ 2 2 First we note that σ1−1 ≈ ∆−1 3 σ1 σ2 and so q = 1 and α = σ1 σ2 σ1 σ2 . Then we have D0 (α′ ) = {σ1 σ1 σ2 σ2 σ1 σ2 } D1 (α′ ) = {σ1 σ1 σ2 σ1 σ2 σ1 } D2 (α′ ) = {σ1 σ2 σ1 σ2 σ2 σ1 , σ1 σ1 σ1 σ2 σ1 σ1 } D3 (α′ ) = {σ2 σ1 σ2 σ2 σ2 σ1 } D4 (α′ ) = ∅ (78) (79) (80) (81) (82) The word with maximal p is σ1 σ2 σ1 σ2 σ2 σ1 for which p = 1. Thus α′′ = σ2 σ2 σ1 . Furthermore D(α′′ ) = {σ2 σ2 σ1 } and we have α′′′ = σ2 σ2 σ1 . Finally we put this together to obtain αG = σ2 σ2 σ1 . The word problem was first solved by Artin [17] and then by Garside [94]. Both of these solutions were algorithmic with exponential complexity. As mentioned above, Garside’s algorithm can be made polynomial time due to a new algorithm by Jacquemard [108] to extract braids. The word problem for braids is important enough for many people to have studied it after Garside. The most efficient algorithm (linear in n and quadratic in L) is given in [31]. Below we present a new algorithm based on rewriting systems. It is not as efficient as the best algorithm for the word problem but it is easily generalizable to the conjugacy case and it is very simple to apply. Because of these features, we regard it as a competitive algorithm. Braids, Knots and Applications 53 4.2 The Word Problem II: Rewriting Systems In this section we will develop a new algorithm based on rewriting systems which are used as a tool in theoretical computer science. We begin with a finite alphabet of constants A and a finite set of variables X . A term t is a finite ordered sequence of constants and variables t = a1 a2 · · · an with n ≥ 0 (i.e. empty terms are allowed) and ai ∈ A ∪ X . A word w is a finite ordered sequence of constants w = b1 b2 · · · bm with m ≥ 0 and bi ∈ A. A substitution ρ for a term t is a map which assigns a word to each variable in t; the resultant word is denoted by ρt. A term rewriting system (TRS) R = {(li , ri )} is a set of ordered pairs of terms li and ri . Each ordered pair in R is referred to as a rule or rewrite rule and is often written in the form li → ri ; the whole TRS is sometimes denoted by →R . A TRS R = {(li , ri )} is applied to a word w0 by determining if w0 contains the word ρli , for some substitution ρ, as a subword. If and only if w0 contains ρli is ρli replaced by ρri . If → is a rewrite rule, then ← is its inverse, ↔ is its symmetric closure (← ∪ →) and →⋆ is its reflexive-transitive closure (→ ◦ → ◦ · · · ◦ →). Two terms t and s are said to be joinable if there exists a term r such that t →⋆ r ←⋆ s. Any li is called a redex and any ri is called a reduct (these are abbreviations of reducible expression and reduced term). A word w0 is thus rewritten into a word w1 if and only if R may be applied to w0 . We may generate a rewrite chain of words w0 →R w1 →R · · · in this manner. R terminates if and only if there exists no rewrite chain of infinite length. R is locally confluent if and only if any local divergence ← ◦ → is contained in the joinability relation →⋆ ◦ ←⋆ . R is confluent if and only if any divergence ←⋆ ◦ →⋆ is contained in the joinability relation →⋆ ◦ ←⋆ . R is complete if it is confluent and terminates. If R is complete a unique normal form exists for each word [19]; the final form obtained by applying R to the word a maximum number of times. It should be noted that the computational power of term rewriting systems is identical to that of Turing machines, i.e. one may be simulated by the other [192]. According to the Church-Turing thesis [59], this means that any function which may reasonably be termed computable is computable using a TRS. It was proven by Birkhoff [40] that the symmetric-reflexive-transitive closure ↔⋆R of a TRS R = {(li , ri )} is equivalent to the set of equations E = {li = ri }. It is an obvious corollary to Birkhoff’s theorem that if there exists a complete TRS R over the alphabet A = {fi , fi−1 } for which ↔⋆R contains exactly the equations {E, fi fi−1 = e, fi−1 fi = e}, then R solves the word problem for the group G = {fi }, E. Note that R also solves the word problem for the monoid associated with G, i.e. the monoid obtained when the inverses of the generators are added to the set of generators and the fact that the generators and inverses are in fact inverses (fi fi−1 = e, fi−1 fi = e) added to the set of equations. 54 Patrick D. Bangert Termination It is, in general, undecidable whether a TRS terminates or not [107]. Since any Turing machine can be modeled using a TRS this is essentially due to the undecidability of whether a Turing machine will stop, the Turing Halting Problem [196]. It is however decidable for a TRS without any variables [66]. Thus, in general, a termination proof is specific to a particular TRS and must be given for it. A common strategy for proving termination is to use a reduction order on the symbols involved in the TRS. We define a reduction order <o as a strict order over the alphabet and variables of the TRS which satisfies: 1. compatibility: For all terms u, v for which u <o v, we have xuy <o xvy for any terms x and y. 2. closure: For all terms u, v for which u <o v and all substitutions σ, we have σu <o σv. 3. basis: <o is well-founded, i.e. there exists a simplest term under <o . If one can show that every possible rewriting operation simplifies any term with respect to such an ordering, then the TRS terminates [65]. Furthermore, a TRS R terminates if and only if there exists a reduction order <o which satisfies ri <o li for every rule li → ri ∈ R [19]. This is true because every step of the rewriting process simplifies the term and there exists a simplest term. Another useful result is that a TRS terminates if and only if it terminates for all instances of its redexes [67]. Some conditions under which the union of two terminating TRSs is terminating are analyzed in [67]. Confluence Like termination, confluence is, in general, undecidable [19]. However, for terminating systems there exists a mechanizable method for deciding confluence [105] that rests on Newman’s Lemma which states that a terminating TRS is confluent if and only if it is locally confluent [163] (we shall prove a generalization of this in lemma 4.13). Local confluence can be decided by a systematic method which searches for critical pairs in the TRS. The concept of critical pairs is difficult to trace in history; for an attempt at a historical survey see [45] and for a good technical treatment see [105]. Given a TRS R = {(li , ri )}, an overlap is a word w = abc such that ab = ρli and bc = ηlj for some words a, b and c, two (possibly equal) integers i and j and substitutions ρ and η. Clearly the overlap abc may be rewritten to both ρri c and aηrj . An overlap is non-critical if the reducts are joinable, ρri c ↔⋆R aηrj and critical otherwise. A critical pair is the (unordered) pair (ρri c, aηrj ) which arises from a critical overlap. It is obvious that if R contains critical pairs, it can not be confluent. The fact that the non-existence of critical pairs is both a necessary and sufficient condition for local confluence is called the Critical Pair Lemma [113]. Later we shall prove lemma 4.14 which contains the Critical Pair Lemma. Completeness If we can find a reduction order for a TRS R, thereby prove termination and find that there are no critical pairs, R is complete and thus Braids, Knots and Applications 55 solves the word problem for ↔⋆R . A general procedure for what to do when we can not do this is called Knuth-Bendix completion from their seminal paper [124]. Again a historical account of this procedure is tangled and [45] is an attempt to unravel it. We shall follow the common practice to call it Knuth-Bendix completion even though, by their own admittion, the initial idea was not theirs. Suppose we have a set of equations E on an alphabet A and a total order <A (this is a reduction order) on A. Construct a TRS R from E by creating a rule li → ri in R from the equation li = ri in E for all equations in E such that the rules are ordered such that li >A ri . Now ↔⋆R is equivalent to E and each rule represents a simplification in terms of <A . Clearly there exists a simplest word, the empty word, and so R terminates. If there are no critical pairs, R is locally confluent and thus complete. If there are critical pairs, order them with respect to <A and append them to R as new rules. Termination still holds and so we continue this process. We may delete duplicate or redundant rules from R between the steps of this method to obtain a smaller TRS. If this method terminates, we have a complete system [124] [106]. If it does not terminate, a complete system may still exist which contains an infinite number of rules. It is possible to collect an infinite number of rules into a finite number of rules by introducing variables. The Knuth-Bendix algorithm has been implemented by several people and can be used to determine, in some cases, whether a complete system exists. The CiME implementation was used for this thesis [138]. Producing rules with variables and proving the non-existence of critical pairs is, at present, beyond the computer implementations and must be done manually. The process described here is simplified; there are more pitfalls, in general, and the method has been considerably extended to take into account many other features (many relevant references are in [45]). The method as described is enough for our purposes here however and is generally enough for a word problem in a finitely presented group. Having reviewed TRS’s, we are now in a position to find a TRS for the word problem in Bn . The braid group Bn is defined formally as Bn =  {σ1 , σ2 , · · · , σn−1 } : σi σi+1 σi = σi+1 σi σi+1 ; σi σj = σj σi for |i − j| > 1 (83) Given a finitely presented group G = X,%E, we can define an associated $ monoid M (G) = X ∪ X −1 , E ∪ aa−1 = 1 for any a ∈ X. It is clear that the equivalence and conjugacy classes of the group G and the monoid M (G) are identical. In order to solve the word problem for Bn , we augment the monoid M (Bn ) with the generator of the center of Bn , ∆2n to form the monoid 56 Patrick D. Bangert ±1 ±2 ±2 , ∆±2 M + (Bn ) =  {σ1±1 , σ2±1 , · · · , σn−1 n } : ∆n σi = σi ∆n ; ±1 ∓1 ∓2 ∆±2 = e; n ∆n = σi σi ±1/∓1 σi±1 σj ±1/∓1 ±1 σi for |i ±1 ±1 ±1 σi+1 σi σi+1  = σj ±1 ±1 σi±1 σi+1 σi = − j| > 1; (84) It is obvious from the definition of the monoid M + (Bn ) that a solution of its word and conjugacy problems provides a solution for the word and conjugacy problemsiword problemiconjugacy problem in the group Bn . We will use Knuth-Bendix completioniKnuth-Bendix completion upon the oriented rules of M + (Bn ) under the reduction order <b −1 −1 −1 ∆2n <b ∆−2 n <b σ1 <b σ2 <b · · · <b σn−1 <b σ1 <b σ2 <b · · · <b σn−1 (85) In practice, this process is laborious and would occupy prodigious space if described in detail. For this reason, we will simply state the result and prove it to be correct. For what follows, we shall represent a braid of the form ∆2k n P as the pair (k, P ). The reason for this is to effectively remove from the braid, in the process of rewriting, any subbraid which lies in the center of the braid group Bn . The reason for this will become apparent when we extend our solution to the conjugacy problem. Removing any ∆2k n from any part of a braid can be done without loss of information because ∆2k n is the generator of the center of Bn and thus its position is irrelevant. By Knuth-Bendix completioniKnuthBendix completion and the necessary manual labor, we obtain the following rewriting system. Wn = { (1) σi−1 → i−1  [dj,1 a1,j ] di,1 a1,i−1 j=1 n−1  j=i+1 (2) σi σj → σj σi for j < i − 1; [dj,1 a1,j ] & k → k − 1; (3) σi σi−1 P σi → σi−1 σi σi−1 P ; (4) σi σi−1 Qσi−1 Rdi,j → σi−1 σi σi−1 Qdi−1,j σi R+ for j < i; (5) n−1  i=1 di,1 a1,i Si → n−1  i=1 Si & k → k + 1 } (86) The variables P , Q, R and Si are (possibly empty) words in the generators σk (and not their inverses σk−1 ) subject to the restriction that the highest generator index k is i − 2, i − 2, i − 1 and i respectively and the lowest generator index in R is j, where i and j refer to the values of the generator indices of the respective rules. The word R+ is obtained from R by increasing all generator indices in R by one. Note that rules 1 and 5 require two replacements to be made simultaneously. A similar, unpublished TRS was Braids, Knots and Applications 57 also found using Knuth-Bendix completion by Yoder [220]. Rules 1 and 5 are simple to understand; the other rules are illustrated in figure 29. Rule 2: Rule 3: Rule 4: Fig. 29. Rules 2, 3 and 4 of TRS Wn illustrated. Theorem 4.5. Wn is complete and solves the word problemiword problem for Bn . Proof. It can be checked easily but laboriously that the system terminates, there are no critical pairs and that its symmetric-transitive closure is the monoid we began with which proves the theorem. 58 Patrick D. Bangert The rules of a TRS are to be applied in a non-deterministic way and a complete TRS always reaches the unique normal form no matter what strategy of rule application is chosen [19]. Since Wn is complete and all strategies are equivalent, we will choose the following strategy. Algorithm 4.6 Input: A word w ∈ Bn . Output: A word w′ ∈ Bn which is the unique representative of the equivalence class of w. 1. Apply rule 1 of Wn as many times as possible. 2. Apply rules 2, 3, 4 and 5 of Wn as many times as possible in order proceeding to the next rule only if the current can no longer be applied. 3. Loop step 2 until no rule may be applied to the word at all. In this case w′ has been found. It is clear that algorithm 4.6 solves the word problem from the completeness of Wn and the fact that once rule 1 is applied as many times as possible, it can not be applied again no matter what other rewrite steps follow as there will be no more inverse generators. From this algorithm, we are able to deduce the computational complexityicomplexity of this word problemiword problem solution. Theorem 4.7. Wn solves the word problemiword problem for any word w ∈ 
Bn of word length l with complexity O l2 n4 . Proof. This can be checked easily by counting how many times the rules are used. 4.3 The Conjugacy Problem I: Garside’s Solution Extending the word problem solution of Garside to the conjugacy case is not hard. First we define two new terms. Definition 4.8. If a positive word b = if for two braid words i and f such that 1 ≤ L(i) ≤ L(b), then i is called an initial route of b and f an associated final route. Note that the definition requires exact equality (=) and not group theoretic equivalence (≈). We begin by constructing all possible initial routes of ∆n . This is easily done using D(∆n ). Replace all of these routes by the base of their diagrams and delete duplicates. This is the set of initial routes of ∆n and denoted by I (∆n ). Exercise 4.9. Compute the set of initial routes of ∆3 . Solution: {σ1 , σ2 , σ1 σ2 , σ2 σ1 , σ1 σ2 σ1 } (87) Braids, Knots and Applications 59 We will now form a set S(β) for a braid word β called the summit set of the braid. This set is the analogue of the Cayley diagram, which we used for the word problem, for the conjugacy problem. We construct the summit set as follows. First construct the Garside normal form of β, i.e. βG , and note its exponent. The set S1 (β) is then constructed by conjugating βG by each word of I (∆n ), computing the Garside normal form of the result and deleting all those words of lower exponent than βG . Iteratively, the set Si (β) is obtained from the set Si−1 (β) by conjugating each element of Si−1 (β) by each word of I (∆n ), computing the Garside normal form of the result and deleting all those words of lower exponent than βG . Suppose the Garside exponent of βG is m and its exponent sum is s. Every word in any Si (β) has exponent sum s and Garside exponent p ≥ m by definition. The Garside remainder of βG is βG and so we have
 s = L βG − mL (∆n ) (88)
 L βG − s (89) m= L (∆n )
 s − L βG p≤ (90) L (∆n ) but we also have p ≥ m and so there are only finitely many p that satisfy the requirements. Hence there are only finitely many distinct words with Garside exponent p and exponent sum s. Thus the process of constructing the sets Si (β) terminates, i.e. there exists a finite integer j such that Sj (β) = Sj+1 (β). Definition 4.10. We define the summit set S(β) to consist of all the elements of Sj (β) which have maximal Garside exponent. The summit exponent t is the Garside exponent of all these braid words and the summit remainder is the Garside remainder βS of the unique element in S(β) with smallest tail. The summit form of β is then βS = ∆tn βS . We state, without proof, the main result of Garside. Theorem 4.11 (Garside’s Conjugacy Theorem [94]). Two braid words β1 , β2 ∈ Bn are conjugate if and only if their summit forms are identical. Proof. For a proof, see [94, 30]. Like the word problem, the conjugacy problem is very important for braid and knot theory and so has received much attention. The first solution was produced by Garside [94] and the best algorithms are given in [31, 84]. We should remark that Jacquemard [108] has used his extraction algorithm to obtain good practical results for small n. All these algorithms still require an exponential amount of computation time as a function of n and L. The important question of whether conjugacy is solvable in polynomial time is solved positively in the next section. 60 Patrick D. Bangert 4.4 The Conjugacy Problem II: Rewriting Systems Conjugacy in Free Groups Suppose that G = Fn the free group of rank n. This group is generated by n elements {fi } for 1 ≤ i ≤ n and no relations [110]. A general word w ∈ Fn takes the form w = fsp11 fsp22 · · · fspmm , 1 ≤ sk ≤ n (91) Since there are no relations in Fn , the word w is unique over its equivalence class if and only if si = si+1 for all i. This condition is trivially obtained from any word w ∈ Fn by applying the (obviously) complete rewriting system Rw (Fn ) = {fsp fsq → fsp+q , ∀1 ≤ s ≤ n} (92) Thus Rw (Fn ) solves the word problem in any free group Fn . Moreover, it does so in a time proportional to the length of the word w. Consider now the conjugacy problem in Fn . We define the ith cyclic permutation C i (w) of a word w in the general form of equation (91) by p′ p′′ m−1 pm p1 p2 C i (w) = fsjj · · · fspm−1 fsm fs1 fs2 · · · fsjj such that p′j + m  (93) pk = i (94) k=j+1 Intuitively, the ith cyclic permutation is obtained by taking the last i generators in the word w and moving them to the front of the word w one by one. We shall say that two words w and w′ are cyclicly permutable (denoted ≈cp ) if and only if there exist an i such that C i (w) ≈ w′ . It is obvious that cyclic permutability forms an equivalence relation for any group G. Proposition 4.12. For any group G, the equivalence relation of cyclic permutability (≈cp ) is identical to that of conjugacy (≈c ). Proof. Any group G has a presentation which may be obtained from some free group Fn of rank n by adding relations [62]. Moreover, if the conjugacy problem is solvable in one representation, it is solvable in all [147]. Suppose w ≈cp w′ , then there exists an i for which p′′ p′ m−1 pm p1 p2 w′ ≈ C i (w) = fsjj · · · fspm−1 fsm fs1 fs2 · · · fsjj where p′j + m  pk = i (95) (96) k=j+1 Let p′′ γ = fsp11 fsp22 · · · fsjj (97) Braids, Knots and Applications 61 Then p′ m−1 pm w′ ≈ fsjj · · · fspm−1 fsm γ ≈ p′ γ −1 γfsjj ≈γ −1 wγ m−1 pm · · · fspm−1 fsm γ (98) (99) (100) Thus we have w ≈c w′ . Now suppose w ≈c w′ , then there exists a γ such that w′ ≈ γ −1 wγ (101) If the word length of γ is L(γ), then we have C L(γ) (w′ ) ≈ γγ −1 w ≈ w Thus w ≈cp w′ . (102) ✷ Fig. 30. The word w given in equation (91) bent into a circle. While the circularity removes the notions of beginning and end of a word, it preserves the directionality of it. We will refer to the set of words which contains the word w and all its cyclic permutations as the cyclic word c(w). If L(w) = m, then this set contains |c(w)| = m elements. Given two cyclic words c(w) and c(w′ ) we test their equivalence by attempting to construct an isomorphism ι : c(w) → c(w′ ) such that ι(a) = a for all a ∈ c(w). Clearly |c(w)| = |c(w′ )| is a necessary condition for the existence of ι. If and only if ι exists, the cyclic words are considered equal, c(w) = c(w′ ). If and only if c(w) = c(w′ ), we have w ≈c w′ by proposition 4.12. The set c(w) may be visualized as the word w ”made circular” as in figure 30. The existence of ι may be established by testing the members of c(w) for equality with the members of c(w′ ) pairwise in the following manner: Select from c(w) an arbitrary member, a say. Check a for equivalence with 62 Patrick D. Bangert all members of c(w′ ). Clearly, if and only if there exists a b ∈ c(w′ ) such that a = b, an ι exists. Since every word has length m and there are m words in c(w′ ), this comparison will take a time proportional to m2 . Thus it is possible
to test  the equivalence of two cyclic words of length m with complexity O m2 . Rewriting Systems for Cyclic Words We shall call a TRS cyclicly terminating, cyclicly confluent and cyclicly locally confluent if it is respectively terminating, confluent and locally confluent under application to all cyclic words over the alphabet of the TRS. It is obvious from the above discussion that a cyclicly complete TRS solves the conjugacy problem. For this reason it is important to develop results about cyclic completeness along the lines of the results for linear words in order to obtain a conjugacy solution. Termination in Cyclic Rewriting Systems We have seen that a TRS R terminates if and only if a reduction order exists [19]. In what follows, we shall assume that this reduction order is a total order; note that this is a stricter requirement than that of a reduction order. Suppose that the alphabet of R is A = {fi } for 1 ≤ i ≤ p. By assumption, p is finite. Consider the total order <R defined by fi <R fi+1 for all i. This can be done without loss of generality as a mapping from A to itself can change the order. Recall that R terminates if and only if ri <R li for every rule li → ri ∈ R. We introduce an integer valued weight metric function g(w) and an integer valued length metric function L(w) on the set of words w written on the alphabet A. The metrics satisfy g (fa1 fa2 · · · fam ) = g (fa1 ) + g (fa2 ) + · · · + g (fam ) L (fa1 fa2 · · · fam ) = L (fa1 ) + L (fa2 ) + · · · + L (fam ) L (fi ) = 1 g (fi ) < g (fi+1 ) (103) (104) (105) (106) We shall call a rule length reducing if L(ri ) < L(li ) and weight reducing if g(ri ) < g(li ). Any rule is a c-obstruction (for commutation-obstruction) if and only if it keeps constant both length and weight. That is, it is a rule which changes the position of the letters only. A c-obstruction obstructs cyclic termination as there exist cyclic words which would give rise to an infinite rewriting chain due the changing of relative position of subwords by the c-obstruction. An example is the cyclic word c(αβαβ) under the TRS R = {αβ → βα}. The rewriting chain will loop between the two states c(αβαβ) and c(βααβ); the period of the loop may, in general, be arbitrarily large. Such looping may be dealt with in two ways. Firstly, one may compare each new cyclic word with the entirety of the rewrite chain so far enumerated. If equality is found, looping has been detected and one may stop. Secondly, one may determine if a subword of the current word commutes with the rest of the word. If this can be determined Braids, Knots and Applications 63 and such a subword is found, the subword may be extracted from the word and the two words should then be rewritten separately. The first method is computationally expensive and does not produce a unique normal form as we would have to consider the entire loop at the end of the rewrite chain as the identifying set of the word. The second method is not necessarily applicable but if it is, it will terminate in a set of subwords which uniquely identify the word. The advantage of the second method over the first is that the number of elements in the set has an upper bound. The braid groups, as we have seen, have non-trivial center. In fact the generator of the center contains every generator. This fact makes it possible to rewrite any inverse generator in terms of inverses of the generator of the center multiplied by generators. As in the word problem case, we shall remove the generators of the center from the word and thus this replacement rewrites the entire braid word with which we shall work in terms of generators only. For the splitting of words to get rid of c-obstructions to fail we need to be in a situation in which we have a word abc such that abc ≈ cab, ab ≈ ba, ac ≈ ca, bc ≈ cb. This can only occur if there is cancellation between a and b in the word ab but this can not happen if there are no inverse generators. This proves that in groups in which inverse generators may be replaced by inverses of elements of the center and generators, this method of overcoming c-obstructions is valid. We conjecture that a TRS R cyclicly terminates if and only if it terminates and contains no c-obstructions or contains c-obstructions that can be removed in the above way. Confluence in Cyclic Rewriting Systems Newman’s Lemma [163] extends easily to the cyclic case as we show below. Lemma 4.13. A cyclicly terminating TRS R is cyclicly confluent if and only if it is cyclicly locally confluent. Proof. The proof is similar to the standard proof (see [105]) and follows immediately from figure 31. The Critical Pair Lemma states that a TRS is locally confluent if and only if it has no critical pairs. Recall that a critical pair arises from an overlap of two redexes in a word which gives rise to a local divergence of rewriting paths which do not meet again. Given a TRS R = {(li , ri )}, a cyclic overlap is a cyclic word c(w) = c(abcd) such that abc = ρli and cda = ηlj for some words a, b, c and d, two (possibly equal) integers i and j and substitutions ρ and η. The cyclic overlap c(abcd) is rewritten to both c(ρri d) and c(bηrj ). A cyclic overlap is non-critical if the reducts are joinable, c(ρri d) ↔⋆R c(bηrj ) and critical otherwise. A cyclic critical pair is the (unordered) pair of cyclic words (c(ρri d), c(bηrj )) which arises from a cyclic critical overlap. It is obvious that if R contains cyclic critical pairs, it can not be cyclicly confluent. 64 Patrick D. Bangert Fig. 31. The proof of Newman’s Lemma (lemma 4.13) in diagrammatic form. We begin at the top with a local divergence which is rectifiable by assumption and thus by induction any global divergence is also rectifiable. It is because of this diagrammatic proof that Newman’s Lemma is also known as the Diamond Lemma. For example, consider the rewrite system R = {abxba → cxc} over the alphabet A = {a, b, c} and some variables x and y. Clearly R contains the cyclic critical overlap abxbabyb which is to be rewritten into bxbcyc and cxcbyb. This cyclic critical pair may be resolved by noting that if the variable contained between the c letters is less than the other, it is that cyclic word which is to be prefered under the lexicographic order c < b < a. That is, we have to add a conditional rule depending on the relative value of the variables. This global rule must be applied, if applicable, with preference over the ordinary local rule. In this way we have extended Knuth-Bendix completion to the cyclic case; note that all rules added in this procedure are global whereas the usual rules of normal TRS’s are local. We shall now prove the extention of the Critical Pair Lemma for the cyclic case. Lemma 4.14. A TRS R = {(li , ri )} is cyclicly locally confluent if and only if it contains neither critical nor cyclicly critical pairs. Proof. This can easily be checked by going through all possible types of overlap. It should be emphasized that the proof lemma 4.14 does not make any assumptions about the termination of R. So we have a definite method for attempting to find a conjugacy problem solution in terms of rewriting. We shall use the braid groups to give an example of this completion process. The Conjugacy Problem in Bn Consider the TRS Wn of equation 86. We have already shown that Wn is complete and solves the word problem for Bn . We shall find that it is neither cyclicly terminating nor cyclicly locally confluent. By a Knuth-Bendix-like completion process, we are able to obtain Braids, Knots and Applications 65 a system which is cyclicly locally confluent. This system will be cyclicly terminating if all c-obstructions are removed, which is possible. The result will be a cyclicly complete system which thus solves the conjugacy problem in Bn . We find four cyclic critical overlaps and make four rules out of these. Furthermore, the augmented system is found to contain no critical pairs. Consider the TRS Gn below which is understood to contain only global rules for cyclic words, i.e. the entire word has to be matched to redexes in Gn . The restrictions on the variables are identical to those of Wn . The ordering <s is the standard shortlex ordering, i.e. words are sorted lengthwise first and then lexicographically using <b . Gn = { (1) n−1  i=1 (di,1 a1,i Si ) → n−1  Si ; i=1 (2) σi σi−1 P σi σi−1 P ′ → σi−1 σi σi−1 P σi−1 P ′ if P <b P ′ or σi−1 σi σi−1 P ′ σi−1 P if P ′ ≤b P ; (3) σi σi−1 Qσi−1 Rσi σi−1 P → σi−1 Qσi−1 Rσi−1 σi σi−1 P ; (4) σi σi−1 Q1 σi−1 R1 σi σi−1 Q2 σi−1 R2 → (107) σi−1 σi σi−1 Q1 di−1,j σi R1+ Q′2 σi−1 R2 if R1 <s R2 or σi−1 σi σi−1 Q2 di−1,j σi R2+ Q′1 σi−1 R1 if R2 ≤s R1 } As described in the solution to the word problem, we will regard a cyclic word c(w) as a pair c(w) = (k, c(ws )). The first entry is an even integer counting the number of copies of d2n in c(w). The second entry is the rest of the word written in the σi . We shall now present an algorithm for the conjugacy problem in terms of Wn and Gn . We prove, in a set of lemmas, that this algorithm solves the conjugacy problem in Bn . Algorithm 4.15 Input: A cyclic braid word c(w). Output: A set of cyclic braid words which collectively are a unique representative of the conjugacy class of w. 1. Apply rule 1 of Wn as many times as possible. 2. Test if w is splittable, i.e. if it is in the form w = w1 w2 where w1 commutes with w2 . If it is, separate w1 and w2 and treat them separately from now on. If not, do nothing. Note that we are testing the linear word w and not c(w). 3. If applicable, apply any rule in Gn and proceed with step 5. If not continue with the next step. 4. Apply any of rules 2 to 4, in that order of priority, of Wn exactly once to each of the separated cyclic braid words, if possible. 5. Go back to step 2 of the algorithm and continue until there is not braid word which may be split further and no braid word to which any of the rules in Wn and Gn are applicable. 66 Patrick D. Bangert 6. The number k and the set of split braid words are now collectively the unique representative required. We note that because of the restrictions on the variable Sn−1 , rule 5 of Wn and rule 1 of Gn are identical. It is obvious from the algorithm that if it cyclicly terminates and Wn ∪ Gn is cyclicly confluent, then the conjugacy problem in Bn is solved by it. Note that the splitting is valid as we have made the replacement of every inverse generator by inverses of elements of the center and generators in step 1, this replacement is the content of rule 1 of Wn .
 Lemma 4.16. Algorithm 4.15 cyclicly terminates in a time O l5 n11 , where l is the initial word length. Proof. Easily done by checking the necessary number of times all the rules need to be applied. Lemma 4.17. Algorithm 4.15 is cyclicly confluent. Proof. Follows from the above discussions. As the algorithm terminates and is confluent, the conjugacy 
it solves problem in Bn with computational complexity O l5 n11 . Conclusion The method to solve the conjugacy problem in the braid groups can be extended to other groups. As we have seen above, these groups must satisfy the following properties: 1. Finitely presented 2. Solvable conjugacy problem 3. Non-trivial center such that each inverse generator may be replaced by inverses of elements of the center and generators. 4. After the initial rules have been constructed according to the methods described, the extended Knuth-Bendix completion must terminate. The fist three requirements may be determined using known methods. The fourth requirement is operational for this method and could possibly be widened. We can not say, before executing the completion process, whether it will terminate. For the braid groups we have seen that a polynomial time conjugacy algorithm is obtained by our new methods, this is important for braid and knot theory as such an algorithm has been sought for some time and solves the first half of the Markov problem. 4.5 Markov’s Theorem Recall that Markov’s theorem says Braids, Knots and Applications 67 Theorem 4.18 (Markov). Two braids α ∈ Bn and β ∈ Bm have isotopic closures if and only if α can be transformed into β by a finite number of applications of conjugacy and stabilization moves. Corollary 4.19. The closure of the braid α ∈ Bn is isotopic to the unlink of k components if and only if α can be transformed into the trivial braid in Bk by using conjugacy and stabilization moves. A B A-1 Fig. 32. Both conjugacy and stabilization are displayed here. We begin with braid B. Conjugation surrounds B with A and A−1 on opposite sides which clearly cancel due to the closure. Stabilization introduces a simple loop at the bottom right of the braid, adds a new string to the braid and thus increases the braid group index by one. Conjugacy was discussed at length already. Stabilization is the move α ↔ ασn with α ∈ Bn (see figure 32). While the conjugacy move is a move within a particular braid group, the stabilization move connects two adjacent braid groups. Therefore the question of detecting closed braid equivalence turns into a combinatorial question about the infinite family of braid groups. Given a knot, we may produce an equivalent knot by taking any segment and twisting it about an axis in the projection plane by π while keeping the rest of the knot stationary. This procedure corresponds to the zeroth Reidemeister move and adds one crossing to the diagram. Any crossing of this type is called nugatory. If we represent a knot by a closed braid by virtue of Alexander’s theoremiAlexander’s theorem, we may also add such nugatory crossings via a combinatorial move, called the Markov or stabilization move. Stabilizing a braid α ∈ Bn corresponds to the operation α → ασn±1 or its inverse. Clearly stabilization increases or decreases the number of strings in the braid and so represents a move in the family of braid groups as opposed to the conjugacyiconjugacy move and equivalence movesiequivalence move which are contained in a single braid group. MarkoviMarkov stated in 1935 [142] that two closed braids are topologically equivalent if and only if they differ by stabilization and conjugacy 68 Patrick D. Bangert moves (recall that conjugacy contains equivalence). This statement became known as Markov’s theorem and was first proven in [30]. In its original form, Markov’s theorem assumes that the closed braid is embedded in S 3 or R3 , this can however, be generalized to an arbitrary 3-manifold [129]. Markov’s theorem transforms the link isotopy problem to a combinatorial question about braids. If two braids α ∈ Bn and β ∈ Bm (with n and m possibly different) are related by stabilization and conjugacy, they are called Markov equivalent which is denoted by ≈M . The decision problem of whether α ≈M β is called the Markov problem or the algebraic link problem. It is possible to find a single move of which both stabilization and conjugacy are special cases and to formulate, in this way, Markov equivalence in terms of this so called L-move [130]. While this L-moveiL-move is intuitive, it is not obvious whether the problem has been simplified by this reformulation. The first question which arises is whether there exist non-conjugate Markov equivalentiMarkov equivalence braid words in the same braid group, that is whether a solution to the conjugacy problem will solve the Markov problemiMarkov problem. This is negatively resolved by showing that the two 4-braids α = σ1m σ2n σ1p and β = σ1m σ2p σ1n with m, n, p different, odd and at least three in absolute value are not conjugate but Markov equivalent [159]. It might be thought that it should be possible to reduce the number of strings in a closed braid equivalent of the unknot to one. This is true as all equivalent closed braids can be reached from each other via Markov’s theorem but the transition involves, in general, increasing the number of strings before they may be reduced to a single string. In other words, a greedy reduction of strings does not reach the minimum string number, also known as the braid index (not even for the unknot representatives) [156]. It is a practical observation that finding a series of moves to demonstrate the Markov equivalence of two closed braids is very difficult. The difficulty of finding such a sequence has lead Birman to believe that it may be simpler to solve Markov equivalence for two braids representing prime knots. While this may be true, it is not, in general, easy to decide whether a braid represents a prime knot. Schubert [178] proved that the factorization sequence of a composite knot is unique and has found an algorithm [179] which finds it. This algorithm, consequently, is able to decide whether a knot is prime. However, the execution of the algorithm rests on Hemion’s algorithm since it must identify the prime factors of the knot, thus no longer necessitating a solution of the Markov problemiMarkov problem since it already solves the link isotopy problem (albeit impractically so). This also shows that this method of deciding primality is not practical. Birman conjectures that a braid represents a prime knot if and only if it is not conjugate to a split braid. Furthermore, if Birman’s conjecture is true and we were to find an algorithm to decide whether a braid was conjugate to a split braid, we would have to solve the Markov problem for this restricted class of braids. If this could be done, we would have a solution to the Markov problemiMarkov problem Braids, Knots and Applications 69 since every braid could be decomposed into its split components and pairwise tested for non-split Markov equivalence. This would not only resolve isotopy but also give the unique prime knot factorization of the knots. Birman’s conjecture is unproven and there exists no algorithm to test whether a braid is conjugate to a split braid. It is possible, however, to solve the Markov problem for certain quotient groups of the braid groups [39]. Since the wordiword problem and conjugacy problemsiconjugacy problem are contained in the Markov problem, solutions for these are desirable and have been given numerous times as mentioned before. The stabilization moveiMarkov move represents the final hurdle before link isotopy is algorithmically decidable and thus it would be interesting to know when a braid α ∈ Bn+1 is conjugate to a braid γσn±1 where γ contains only the generators σi for 1 ≤ i ≤ n − 1, for then one could reduce α to γ using the Markov move. While this has been done [143], the algorithm depends on Garside’s conjugacy algorithm [94] which has exponential complexity. Moreover, if two braids were reduced in this way to the minimum string numberiminimum string number, they are not, in general, conjugate in this final braid group if they are Markov equivalentiMarkov equivalence and thus this decision procedure does not solve the Markov problem either. We have defined the exponent sum exp(α) of a braid α as the sum of the exponents of the Artin generators of α. It is obvious that the exponent sum is a conjugacy class invariant but not a Markov class invariant because of stabilization. Thus it is possible for two braids to be Markov equivalent and have different exponent sums. In getting from one braid to the other, the exponent sum must be made equal somewhere in the chain of moves; this can clearly only be accomplished using stabilization. Stabilization can increase or decrease the exponent sum depending whether we add σn or σn−1 or remove either of these. It also changes the number of strings. We may think that starting from a positive braid, we should be able to reach any Markov equivalent positive braid by going through a pure positive sequence of braids; that is, we may think that positive Markov equal braids are positively Markov equal. We note that this would only be possible if the difference in exponent sum between the two braids was precisely their difference in number of strings. We conjecture that positive Markov equal braids are not positively Markov equal. Much work was done by Birman and Menasco on various properties of links which could be determined from their closed braid representatives (this work was published in the six-paper series [32], [33], [34], [35], [36] and [37]). They prove that there exists a complete numerical invariant for knots but find this invariant only for knots which are closed 3-braids. The invariant for closed 3-braids is described extensively and can be used to determine the braid index and whether the knot is split, composite, amphicheiral or invertible. They also define a new type of move on braids, the exchange 70 Patrick D. Bangert moveiexchange move, and prove a Markov-like theoremiMarkov’s theorem for it. See [38] for a summary of this work. 4.6 The Minimal Word Problem A well-known problem of combinatorial braid theory is the minimization problem: Given a braid A ∈ Bn find a braid Am such that A ≈ Am and L(Am ) ≤ L(A∗ ) for any braid A∗ ≈ A where L(A) denotes the word length of the braid A. In the Artin representation of Bn , the number of generators required to write down a braid word, its length, is equal to the crossing number of the topological braid. In practice, we find that by moving a few of the strings of the topological braid, its crossing number may be reduced, making the braid simpler. It would be especially useful to possess a general method to compute an equivalent braid of minimum crossing number. Apart from many applications, this problem is well-known in combinatorial braid theory and is of independent mathematical interest. Given a braid A ∈ Bn in the Artin generators, the question whether there exists an equivalent braid A′ ∈ Bn of shorter length has been shown to be NP-CompleteiNP-complete by Paterson and Razborov [168]. Not only does this mean that this question is computationally equivalent to all other NPComplete problems, it also means that (unless P = NP) any algorithm which answers the question would execute in exponential-time in n. Since Paterson and Razborov’s result refers to the minimization problem for general n, we ask whether it is also an NP-complete question for particular n. This question is explicitly asked as open question 9.5.6 on page 209 of [84] and it seems to have been negatively answered in an unpublished preprint by Tatsuoka five years earlier but we were unable to obtain it [191]. In proving the NP-Completeness of the problem, Paterson and Razborov showed that the problem can be reduced to a known NP-Complete problem. This does not however provide a usable algorithm. For 3-braids, a linear complexity algorithm has been found [27] but no general algorithm for n > 3 exists. A minimization algorithm in the band-generator presentation of the braids groups has been found for n = 3, 4 but the length of the braid in this presentation is not equal to the crossing number [218] [114]. It is untypical of a group for which the word problem is solvable that no unique normal form of minimal length in some naturally arising presentation exists for the braid groups. A unique normal form of minimal length in certain natural presentations of free groups, HNN-extensions and free products exists, for example. After a little experimentation, it is clear that a braid must, in general, be increased in length before it may be reduced to minimum length algebraically. We show that a certain readily obtained braid provides an upper bound for this necessary increase in length and prove several properties of this braid. We explicitly construct a set of words which must be searched for a certain Braids, Knots and Applications 71 property in order to obtain a minimal length representative of any braid. This constitutes an algorithm to solve the minimization problem. Since the set of words which must be searched is, in the worst case, exponential in size, the algorithm takes an exponential amount of time to complete. Exercise 4.20. Find a braid which is non-minimal in length and which must be increased in length (by introducing pairs like σi σi−1 ) before it can be shortened to minimal length. An example is the braid σ2 σ1−2 σ2−2 σ1 σ2−1 σ1 σ22 σ1 σ2 . Denote by Am any braid which satisfies Am ≈ A and L(Am ) ≤ L(A∗ ) for all braids A∗ ≈ A. We now prove a basic lemma which connects Amax −s(A) ′ and Am . Recall that Amax = ∆n A where s(A) is the number of inverse ′ generators in A and A is positive. Theorem 4.21. For any braid A, it is possible to obtain Am from Amax by operations which monotonically decrease or keep constant the length of the braid. Proof. By construction Am ≈ Amax ≈ A and Amax and A are at least as long Am . Exponent sum is an equivalence class invariant so that s(Am ) ≤ s(A). Replace each inverse generator in Am with the braid given in proposition 4.1 and then use equation σi ∆n = ∆n σn−i to bring all the fundamental braids to the front to obtain the braid Ammax = ∆n−s(Am ) A′m ≈ ≈ −s(A) But Amax = ∆n we have that m )−s(A) ∆n−s(Am ) ∆s(A ∆ns(A)−s(Am ) A′m n ∆−s(A) ∆ns(A)−s(Am ) A′m n (108) (109) (110) A′ and since the braid groups are left-cancelative [94], ∆ns(A)−s(Am ) A′m ≈ A′ (111) with both words positive. Since positive words are positively equal [94], there exists a sequence of braids Bi for 0 ≤ i ≤ q with B0 = A′ , s(A)−s(Am ) ′ Bq = ∆n Am , Bj and Bj+1 different by a single application of the braid group’s defining relations and Bi positive for all i. Since exponent sum is an equivalence class invariant, L(Bi ) = L(A′ ) for all i. From Amax we may thus reach the form of Ammax in equation (110) keeping the length of the braid constant. From this form, we may reach Am by operations which monotonically decrease or keep constant the length of the braid. Thus there exists a sequence of braids Wi for 1 ≤ i ≤ p with W0 = Amax , Wp = Am , Wj and Wj+1 different by a single application of the braid group’s defining relations and L(Wj+1 ) ≤ L(Wj ), which proves the lemma. ✷ Theorem 4.21 basically establishes that we may reach a minimum length representative from Amax by rearranging and cancelling generators only; it 72 Patrick D. Bangert thus, in principle, removes the difficulty we pointed out in the introduction of occasionally having to increase the length before being able to decrease it to an absolute minimum. We present an extension to the Cayley diagram construction which draws the diagram of any braid word (as opposed to positive braid words only). The diagram is a list of all those braid words which may be obtained from the given word by rearranging only. Algorithm 4.22 Input: A braid word A. Output: A list D(A) of all braid words B which may be obtained from A by rearranging of generators only. 1. Define the diagramiCayley diagram of zeroth order as the set D0 (A) = {A}. 2. The set Di (A) is obtained from the set Di−1 (A) by the following procedure: a) Fix attention on a particular member α of Di−1 (A). We read α from left to right and decide at each position whether we may apply any of the moves in equations (112) to (115). σi σj ↔ σj σi for |i − j| > 1 σi σi+1 σi ↔ σi+1 σi σi+1 σi σi−1 ↔ σi−1 σi σi σi−1 σj ↔ σj σi σi−1 (112) (113) (114) (115) b) If we may, we apply it and store the resultant braid word β in Di (A) if and only if β is not already contained in Dj (A) for 0 ≤ j ≤ i. c) We continue to read across α until we have considered all braid words which may be reached from α by a single application of the moves in equations (112) to (115). d) Apply steps (a) through (c) for every braid in Di−1 (A). If Di (A) = ∅, then the algorithm is done. 3. The diagramiCayley diagram D(A) of A is the union of all the Di (A), & & & D(A) = D0 (A) D1 (A) · · · Dm (A) (116) We show the correctness and termination of this algorithm. Lemma 4.23. Algorithm 4.22 terminates for every A and succeeds in listing all braid words B which may be obtained from A by rearranging of generators only, that is using the braid group relations without introducing or removing any generators. Proof. D0 (A) is, by definition, finite. It is obvious that for any braid word of finite length, the moves in equations (112) to (115) may be applied a finite number of times. Thus, by induction, every Di (A) is finite. The number of distinct braid words of a given finite length is finite and since the Di (A) Braids, Knots and Applications 73 are, by construction, non-overlapping, their union must be finite. Thus there exists an m such that Dm+k (A) = ∅ for every k > 0. Thus the algorithm terminates for every A. The moves listed in equations (112) to (115) exhaust all possibilities allowed in the braid group under the stipulation that no generators must be removed from or introduced into the word. Thus each word which may be reached from A by rearrangement of generators will eventually be reached by algorithm 4.22 and so the algorithm succeeds in listing all the required braid words. ✷ Theorem 4.21 gives the following corollary. Corollary 4.24. D(Amax ) contains a braid of the form EAm for E ≈ e, the identity in Bn . Proof. By construction D(Amax ) contains all braid words equivalent to Amax by rearranging only. By lemma 4.21, Am can be obtained by a sequence of operations which keeps the length constant or decreases it. Each operation which decreases the length does so by eliminating a sub-word like ei = σi σi−1 ≈ σi−1 σi . Since for all i ei ≈ e, the identity in Bn , we have ei σj±1 ≈ σj±1 ei , ei ej ≈ ej ei (117) for any i and j. Let us now agree to construct the aforementioned sequence of words without eliminating the sub-words ei but using equation (117) to bring them all to the left of the word. At the end, we will obtain a word of the form A∗ = EAm where E ≈ e is a braid consisting of all these sub-words ei . The most general form of E is qn−1 (118) E = eq11 eq22 · · · en−1 with qi ≥ 0 for all i. So if we could extract E from Amax , we would, in the process, obtain Am . Since the form EAm is obtained 'n−1by rearrangements only, ✷ L(E) ≤ L(A∗ ) = L(Amax ). This indicates that i=1 2qi ≤ L(Amax ). Given a braid A, we thus find Am by constructing the diagram D(Amax ) and selecting the word with the largest number of cancellation pairs such as σi σi−1 . Clearly there will be more than one braid word for the same number of cancellation pairs. We may agree to choose the least braid word lexicographically for definiteness. It is obvious from the construction that this will be a unique form of minimal length for the braid A. We thus have an algorithm to find Am for any A. It is regrettable that the diagram D(Amax ) is, by construction, very large. Two questions are left to ask: Can we make the result stronger and how large is a typical diagram? In theorem 4.21 we achieved an upper bound for the necessary increase in length of a braid before it may be reduced to a minimum length. One would 74 Patrick D. Bangert like to simplify the result somewhat but we shall show in this section that the two straightforward attempts to simplify or strengthen theorem 4.21 are doomed to failure. First we show that we may not, in general, shorten Amax to the Garside normal form. Lemma 4.25. It is not, in general, possible to obtain Am from G(A), the Garside normal form of A, by operations which monotonically decrease or keep constant the length of the braid. 3 Proof. Consider the braid α = ∆−2 3 ∆3 σ1 . The Garside normal form of α is −1 3 G(α) = ∆3 σ1 and the shortest braid which can be obtained from G(α) by rearranging and cancelling only is α′ = σ1−1 σ2−1 σ12 . The original form of α is the same as αmax and we make the following moves on it 3 α = ∆−2 3 ∆3 σ1 ≈ σ2−1 σ1−1 σ2−1 σ1−1 σ2−1 σ2−1 σ2 σ2 ∆3 σ1 −2 2 ≈ σ2−1 ∆−1 3 σ2 σ2 ∆3 σ1 ≈ σ2−1 σ1 (119) (120) (121) (122) which is shorter than α′ and is in fact the minimal length of this 3-braid. This provides an example for which the minimal length is not obtainable from the Garside normal form of the braid by rearranging and cancelling only and thus proves the lemma. ✷ One may think that it would be sufficient to list the diagram of the negative and positive sub-braids of Amax and search for a maximal length subbraid which is common to the end of the first and the beginning of the second diagram but this is not true as the following lemma shows. Lemma 4.26. There does not exist an Am in the form A1 A2 with A1 negative and A2 positive for every A. Proof. Consider the braid A = σ1−1 σ2 σ1−1 , the Garside normal form of which is G(A) = ∆3−2 σ2 σ1 σ1 σ1 σ2 . In fact, A is already minimal as can be seen by Berger’s algorithmiBerger’s algorithm [27] or by using the above procedures. We wish to show that there does not exist another braid equivalent to A of length three in the form A1 A2 with A1 negative and A2 positive. Since exponent sum is a conjugacy class invariant, we need only check eight cases. Below we list the eight 3-braids of length three and exponent sum -1 in the required form and their Garside normal forms. Braids, Knots and Applications σ1−1 σ1−1 σ1 → ∆−1 3 σ1 σ2 −1 −1 −2 σ1 σ1 σ2 → ∆3 σ2 σ1 σ1 σ2 σ2 σ1−1 σ2−1 σ1 → ∆−1 3 σ1 σ1 −1 −1 −1 σ1 σ2 σ2 → ∆3 σ1 σ2 σ2−1 σ1−1 σ1 → ∆−1 3 σ2 σ1 σ2−1 σ1−1 σ2 → ∆−1 3 σ2 σ2 σ2−1 σ2−1 σ1 → ∆−2 3 σ1 σ2 σ2 σ1 σ1 σ2−1 σ2−1 σ2 → ∆−1 3 σ2 σ1 75 (123) (124) (125) (126) (127) (128) (129) (130) Since the Garside normal form solves the word problem and none of the above Garside normal forms are identical to G(A), the braid A does not possess a minimal length representative in the required form. ✷ Let a be a n-braid of length L with diagram D(a). Consider the braid a′ = aσi σi−1 for some 1 ≤ i < n. We are concerned with the size of D(a′ ) in terms of the size of D(a). For each member of D(a), the cancellation pair σi σi−1 may appear in any place in both possible orders (σi σi−1 and σi−1 σi ), so in 2(L+1) positions. There may be further moves possible by use of the braid group relations but the number of these are clearly bounded by a function linear in L. So the diagram of a word will increase in size by a factor linear in its length for each possible cancellation pair. Given a random positive n-braid a of length L, how many members will D(a) have, on average? We conjecture that: Conjecture 4.27. For any braid a ∈ Bn of length L, we have that |D(a)| ≤ |D(∆pn )| with p = ⌈2L/(n(n − 1))⌉. Conjecture 4.27 would provide an upper bound for the size of the diagram of any word in terms of the diagrams of the diagrams of ∆pn which topologically are a series of p half-twists of the braid strings about the vertical axis. In extensive computer simulations, the conjecture was checked and seems to hold. What it seems to indicate is that the half-twist has the most topological freedom for its length and number of strings under the constraint that the crossing number must be kept constant. This is quite intuitive, yet the conjecture seems to be difficult to prove. We have investigated the diagrams of several ∆pn for their size and for the distribution of braids over the sub-diagrams at each stage of the construction in algorithm 4.22. In table 1 we list the size of the diagram and maximal sub-diagram index for p half-twists on n strings. We conclude that the diagram of a typical braid word grows exponentially with its length and braid index and thus our method of finding the minimal length braid word equivalent to a given braid has exponential complexity. This is not surprising as the problem is NP-Complete. We shall give a heuristic algorithm and other methods later. The properties of the braid 76 Patrick D. Bangert Table 1. The Size of Diagrams of Fundamental Words n 3 3 3 3 3 3 3 4 4 5 p |D (∆pn )| max. i 1 2 1 2 8 2 3 38 5 4 196 8 5 1062 13 6 5948 18 7 34120 25 1 16 7 2 1654 15 1 768 25 groups that made the above solution possible are: (i) It is possible to write all inverse generators as products of the generator of the center and a positive word, (ii) the defining relations relate positive words only and (iii) the braid groups are right and left-cancellative. It is likely that any group which has these properties, has an analog of the Garside normal form and has a solution to the minimum word problemiminimum word problem similar to the one above. Solving the problem exactly is an expensive endeavor and so we ask for approximate methods. It turns out that magnetic relaxation is an important application of this problem and gives rise to good methods to solve it. We shall delay the discussion of these to section 5.3. It is possible to solve the problem heuristically using a purely algebraic algorithm which we now present. Recall that the braid group Bn is defined by Bn =  {σi } : 1 ≤ i < n; σi σj = σj σi |i − j| > 1; σi σi+1 σi = σi+1 σi σi+1  . (131) (132) An n-braid A of c crossings is a word in Bn of word-length c, so the general form of A is A = σaǫ11 σaǫ22 · · · σaǫcc ǫk = ±1, 1 ≤ ak < n, ∀k : 1 ≤ k ≤ c. (133) Consider an n-braid A of the form given in equation 133. Suppose we wish to find the n-braid Am equivalent to A such that the length L(Am ) of Am is minimal over the equivalence class of A. It has been shown [168] that this question is NP-completeiNP-complete and hence computationally difficult (if P = NP, it is intractable). The following presents a heuristic algorithmiheuristic algorithm for getting close to Am . We begin with the leftmost generator of A and attempt to move it to the right using both braid group operations. If we can cancel it along the way, we do and if we can not, we move it back to where it started. In this way, we proceed to move all the generators as far to the right as possible. Then we begin at the end and Braids, Knots and Applications 77 move each generator as far to the left as possible in the same manner. This algorithm will always produce an equivalent braid A′ such that L(A′ ) ≤ L(A). We consider L(A) generators and
move them O(L(A)) moves to the right and  left. Thus this algorithm takes O L(A)2 time and constant memory. In fact we move a particular generator at most L(A) generators and this is only for the case when all the other generators commute with it, thus the average case complexity is likely to be close to linear in L(A). This algorithm will not produce a minimum length representative in all cases because it can not unravel complex crossings. To get to the minimum length would require more subtle transformations than just movements to the right or left, which topologically correspond to pulling the strings apart from underneath the crossing. However, as computer experiments show, it does do quite well. Let us calculate an upper bound to the reduction ratioireduction ratio obtained by this method as a function of n and c. To calculate these, consider the likelihood that a particular generator will be followed by its inverse, which is just Q0 = 1/2(n − 1). The probability Qj that a generator and its inverse are separated by j generators through which either can be moved is the corresponding probability for j = 1 to the power j. We require the number of braids of length 1 which may be generated so as not to contain the generator interfering with the movement of generator σi . If i = 1 or n−1, this is 2(n−3) and 2(n − 4) otherwise. Thus  Qj =  = * 2(n − 4) ( 2(n−1)−2 2(n−1) n2 − 5n + 5 (n − 1)2 ) + 2(n − 3) 2(n − 1) +j ( 2 2(n−1) ) j  Q0 Q0 (134) (135) The final factor of Q0 is present because the generator after the sequence of j generators is required to be inverse of the original generator, an event with probability Q0 . To get the total probability Q of being able to cancel a generator σi with its inverse by simple exchange movements over the length j = 0, 1, · · · , we must sum these probabilities in order weighted by the probability that their predecessors did not happen. Thus Q = Q0 + (1 − Q0 )Q1 + · · · + j−1  (1 − Qk )Qj + . . . (136) k=0 Note that since the exchange move is not allowed for n = 3, Q = Q0 for n = 3. The reduction ratioireduction ratio R which occurs as a consequence of this probability is R = 1 − 2Q since each time that the event happens two generators may be canceled. Note that in this calculation we have considered the probability that a generator can be moved next to its inverse in the word 78 Patrick D. Bangert using only the far commutation relationifar commutation that σi σj = σj σi for |i−j| > 1 in a long braid. The heuristic algorithm however uses both braid group moves to attempt to move generators next to their inverses. Thus R is an upper bound for the reduction ratio achieved by the heuristic algorithm as the braid becomes long. In §6 we present the results of the algebraic reduction of a large number of braids but a few comments about the efficiency of the algorithm are in order. The only exact algorithm to minimize braidiBerger’s algorithm is valid only for n ≤ 3 [27] and by comparing this heuristic to this exact algorithm, we find that the heuristic finds a braid the length of which is within five percent of the length found by the exact algorithm and that it reaches the actual minimum in 0.005 of all cases. This shows that the heuristic is quite effective for n = 3 (note that reduction for n = 1, 2 is trivial since B1 , B2 are free groups). 5 Applications of Braid Theory In the discussion of figure 4 in section 1, we observed that the braid under discussion was simpler than the geometry of the dynamical system that gave rise to it. It is frequently possible to gain understanding of some physical mechanism by ignoring the details and focusing on topological structures. In this section, we will give some examples of systems in which this approach has been useful. For reasons of space, we have made no attempt to be exhaustive. The number and depth of the applications of braid theory go far beyond the treatment here. 5.1 Topology as a Conserved Quantity in Classical Dynamics Physics, to a large degree, is a search for conserved quantities. The fact that the total energy of a physical system remains constant over time can explain a plethora of phenomena. The conservation of angular momentum is essential to our understanding of quantum physics and the atomic and subatomic worlds. The details of a system are in constant state of change but there are certain properties that remain the same. It is these that improve our understanding on the fundamental level. When we look at a system, we observe sizes, lengths, locations and so on all of which will change very shortly. It is frequently something unobserved that stays constant and many times it is the shape of the overall system or the shape of some abstraction of system properties (for example a phase space trajectory) that is unchanged. Such information is topological in nature and we may use the theory of topology to classify the possibilities. Consider, for example, the dynamical system of n particles moving in a a 2 with c a constant, rij = |xi − xj | the plane relative to potential V = crij distance between particle i and j and a the parameter of the system. We will Braids, Knots and Applications 79 trace the evolution of the system in three dimensions with the vertical being time. As such, the n particles will build an n-braid over time. If the particles do not move (c = 0), the braid is the identity braid e ∈ Bn . Consider an n-braid embedded into R3 . The question is whether there exists a periodic orbit of the dynamical system of n particles moving under the potential V . It can be shown that if a ≤ −2 all braid types have periodic orbits but as a increases above -2, some no longer have periodic orbits and when a has increased to a = 2, only the braids ∆2n are left over. It is interesting that it is the braids in the center of the braid group that should survive until the end. The investigation of which braid types are and are not possible for a given a was carried out but will not be covered here in detail [151]. For n point masses arranged on a circle and moving under the potential V , the orbit is periodic and gives rise to the topology ∆2n in the obvious way. It is also stable for a > −2 and unstable otherwise [13]. The circle is the simplest knot possible. To obtain more complex knots, we must move to three dimensions and there the braid classification does not work any longer (any knot can be untied in four dimensions). The question that does arise is whether the orbit of n point masses arranged along a knot in three dimensions is stable or not. It is claimed that this is asymptotically stable (as n tends to infinity) for any knot type in which the masses move under the gravitational potential (a = −1) [46]. These configurations are thus new solutions to the n-body problem. However for a finite number of masses, this configuration will, in general, be destroyed after some time. No simulation efforts have gone into this problem and it remains an important open problem to decide whether these orbits are stable enough to occur in nature [176]. Numerical experiments similar to the kind suggested have been done in two-dimensions [183]. So far the particles have just moved according to a very simple dynamical equation (Newton’s law with a central potential V ). If we generalize this dynamics to that of a parabolic partial differential equation, the trajectories may still be classified using braid theoretic methods along the above lines [96]. Ordinarily, topological invariants are calculated for a particular given topological object. We have seen, for example, how to calculate the Alexander and Jones polynomials for a closed braid. Reversing the process is highly non-trivial, i.e. given a polynomial construct a closed braid whose Alexander (or Jones) polynomial is equal to that given one. This reverse construction is particularly difficult for many invariants defined in terms of projections of the knot into the plane. If we had genuinely three dimensional definitions of these invariants, perhaps an inversion would be easier. The theory of Vassiliev invariants calculated via Kontsevich integrals which we shall not review and Witten integrals which we will briefly discuss below gives a three dimensional integral definition of the classical knot invariants [197, 127, 213]. Using these integrals, it is then possible to construct a dynamical system of 80 Patrick D. Bangert particles moving in the plane which (if the time dimension is the upward direction) generates a braid whose closure has the given invariant [28]. The inverse problem can thus be solved. 5.2 Knot Theory in Fluid Dynamics As mentioned in section 2.2, knot theory was created from the idea that atoms could be knotted vortices in the ether, which was thought to act like a fluid. The relationship between knot theory and fluid dynamics therefore goes back to the conception of knots as a mathematical discipline. The practical science of knot tying on the other hand was, for the most part, invented by sailors who can lay claim to more than 90 percent of knots actually used. Fluid dynamics is an important old field of science. Air and water are just two examples of fluids we encounter every day. Improving our understanding of the dynamics of fluids has given us aeroplanes, modern ships and many other types of hydrodynamical technologies all the way to alternative power resources in the oceans, rivers and wind. The basic question of fluid dynamics is what the flow of the fluid, initially prescribed, will look like after it has flowed down some path which possibly contains obstacles. There are a set of differential equations known as the Navier-Stokes equations which describe how the flow profile changes. Unfortunately fluids are very complex and the Navier-Stokes equations are rarely soluble. So complicated are these equations that the Clay Mathematics Institute has offered a millennium prize of one million dollars to the person who proves certain statements about the existence of solutions to these equations, nevermind actually finding them! In view of the lack of analytical solutions, people have adopted several alternative avenues for progress. Some try to solve the equations numerically. Some follow a random selection of fluid molecules down the path in a so called Monte-Carlo simulation. Some try to isolate the idealized cases where an analytic solution does exist. Others hunt for properties that will remain constant over the evolution of the fluid, properties that we call invariants. Many invariants are structural or global in nature and this is where topology is useful. To describe actual invariants we must get more specific about the physical situation we are concerned with. Here, we will give an example which will be of interest in both solar magnetohydrodynamics and DNA recombination. To be specific we will adopt the language of magnetodydrodynamics. Suppose we are faced with a magnetic field B with compact support in ℜ3 (B = 0 outside some compact domain D ⊆ ℜ3 , the domain D is called the support of B.). A magnetic field can be represented as the curl of a vector function called the vector potential A, B = ▽ × A. Definition 5.1. We define the helicity H of the field by , H= A · BdV D (137) Braids, Knots and Applications 81 where the integral is over the support D of B. The helicity is an invariant of the field if the field is “frozen” which means that the field obeys a restricted form of field equations than the full magnetodydrodynamic field equations [214, 215]. While such a restriction is significant mathematically, it is still a realistic model for a variety of physical phenomena. Consider a particular field line K. No field line may have free ends (Maxwell’s equations – no magnetic monopoles) and so K must be closed and as a closed curve in ℜ3 , it is isotopic to some knot including the unknot. Consider the field in a tubular neighborhood T around K. It can be shown [150] that if Φ is the flux of B across perpendicular cross-sections of T , we have (138) H = hΦ2 where h is a number which we shall call the winding number. It is a remarkable result by Cǎlugǎreanu that h is the sum of two geometric properties of K called the writhe W and twist T . Definition 5.2. The writhe W of a knotted field line K is defined by the double integral - (dx × dx′ ) · (x − x′ ) 1 W (K) = (139) 4π K K |x − x′ |3 W (K) varies continuously with deformations of K but tends to the topological definition of the writhe of a knot as K is deformed to lie very nearly in a plane (see the discussion of exercise 2.2). Definition 5.3. The twist T of a knotted field line K is defined by the integral 1 (N′ × N) · tds (140) T (K) = 2π K where s is arc length along K, N(s) is a unit normal from K to some given field line and t is a unit tangent vector to K. The twist is the twist of a ribbon with K and the field line as boundaries. Almost all of the twist arises from the linking of K with the field line. The linking number lk(a, b), which measures the linking between two curves a and b, is defined as the sum of the characteristics ǫ of each crossing between a and b. If the crossing is right-handed ǫ = 1 and if it is left-handed ǫ = −1. These are the same characteristics used in the definition of writhe for a knot, see the discussion after exercise 2.2. We thus have h=T +W (141) The winding number h is an invariant of a frozen-in field but twist and writhe are not. Thus we may convert twist into writhe and vice-versa. The 82 Patrick D. Bangert physical mechanism by which such a conversion might take place has been investigated [136] but may be visualized easily. Take a rubber band and cut it so that you have two ends. While holding one end still, twist the other around its axis many times and tie the ends of the band together again. What you have is an unknot without writhe which is heavily twisted. When you let go of the rubber band, it will partially untwist by folding back on itself globally, that is it will convert some twist into writhe. If you keep track of the appropriate numbers, you may see that the conservation law holds. Equation 141 is known as Calugareanu’s theorem and is very important not only in fluid dynamics but also in DNA research [49, 50, 51]. The example of the rubber band is quite instructive and so we encourage the reader to actually try it. It illustrates a further point. While twisting the band, you have injected energy into the band which it is forced to store as you are still holding it in place. When you let it go, the elastic potential energy converts partially into kinetic energy. The band moves into an energetically more favorable configuration which involves converting some twist into writhe. This process is called relaxation. The main questions about relaxation are: (1) What will the final configuration look like? (2) How much energy will the system loose in relaxing? We will investigate these questions using an example in the next section. 5.3 Braid Relaxation Suppose we have some braid structure b. The total geometrical string length over the whole braid will be called its energy. Initially the braid has energy E and we want to know whether there exists a continuous deformation of b into some configuration b′ isotopic to b with energy E ′ < E. A physically realistic deformation is an elastic relaxation. From this point of view, we regard each string of the braid as an elastic string. As the endpoints are fixed, the lowest energy state is one in which all strings are straight lines. The only way in which all strings can be straight is if b is isotopic to the trivial braid. We thus have a positive lower bound on the energy of a braid and we know that the lower bound can only be reached from exactly one topology, namely the trivial one. If the braid contains σi σi−1 as a subbraid, we may delete it. This deletion reduces the number of crossings by two and also reduces the energy of the braid as both strings can be made shorter. This and similar considerations may lead one to believe that crossing number is a very similar measure of complexity as energy. Topologically an equivalent braid with less crossings is simpler and this elementary observation gave rise to the minimal word problem (see definition 2.31). We are thus lead to ask whether the search for a minimal energy configuration is essentially the same as the search for a minimal crossing number? The unexpected answer is that they are different. In this section, we investigate several different approaches to obtain minimal configurations: we employ three different relaxation techniques and com- Braids, Knots and Applications 83 pare these with each other and with an algebraic heuristic algorithm, in terms of minimization (of energy and crossing number) and time efficiency. By energy we mean total string length of the braid. It is found that more than half of the crossings of a sufficiently large braid (in terms of crossing number and number of strings) are redundant. We analyze the different methods and say in what circumstances which method is to be favored and conclude that minimum braid energy and minimum braid crossing number are substantially different measures of topological complexity for braids. Topological constraints appear in many complex systems. In biology the amount of twisting and knotting of DNAiDNA molecules can affect molecular interactions and dynamics [115]. In polymer physics the degree of entanglement of the polymer filaments helps to determine the elastic properties of the polymeripolymer [15]. In astrophysics, applications involve the behavior of magnetic fieldsimagnetic field (such as those found in stars and accretion disks) with complex topologies [166, 26, 27, 44, 167]. In dynamical systems theory, the time history of the system can be represented by a set of braided particle orbits; the topology of the braid reflects qualitative aspects of the dynamics [57, 43, 144, 151]. In turbulence theory the degree of entanglement of the vortex linesivortex lines provides a statistical measure of flow properties; this measure is distinguished from most others used in turbulence by being based on the flow in real space rather than on the spectral transform of the flow in Fourier space. In statistical mechanics, braid and knot theory has significantly contributed to exactly solvable models via knot polynomialsipolynomials, for example [199]. Random knottingirandom knot, as opposed to the random braidingirandom braid discussed here, has also been investigated [181]. All these applications involve a set of curves (e.g. long molecules, magnetic or vortex lines) which are knotted, linked, or braided. Knot theorists have devoted great effort to classifying such objects. One important part of this effort concerns finding measures of complexity. This idea goes back at least as far as TaitiTait, who first set out tables of different knot types [190]. Tait organized the knot tables according to a simple complexity measure, the minimum crossing number Cmin . This number gives the minimum number of crossings of a knot as seen in any two dimensional projection. There are two types of topological invariants. The first, sometimes called isotopic invariants, involve quantities that remain constant if we deform the set of curves. Examples include the Gauss linking integral, helicity integrals, and knot polynomials. The second type involves quantities which do change when the curves deform, but have a lower bound. This second type of invariant can be regarded as a measure of topological complexity [186] of which both crossing number and energy are examples. This chapter investigates the crossing number and energy for random braidsirandom braid. Braids consist of a set of curves stretching between two parallel planes. The endpoints of the curves are fixed but, between the two 84 Patrick D. Bangert planes, the curves are free to move so long as they do not cross through each other. Braids are important in knot theory because (unlike knots) they can be readily classified using group theory [31]. They are also important in solar physicsisolar physics, as the field lines within a coronal magnetic loop are braided. (In fact, a coronal loop forms an arch with both ends in the photosphere. But a simple geometrical transformation straightens out the arch into a cylinder with ends on two parallel planes.) Energy, of course, has the most immediate physical significance. For example, a solar magnetic loop usually stays close to the energy minimum (equilibriumiequilibrium) state consistent with the field topology. Sometimes this equilibrium becomes unstable; a rapid reconnection event changes the topology and energy is released in a flareisolar flare. Crossing number, on the other hand, relates more directly to the geometry of the field lines. The state of minimum crossing number may not be exactly the minimum energy state, but one may conjecture that they will be close. We can, in fact, find strict lower bounds for the energy of a magnetic field given its crossing number. This has been done for fields in a spherical geometry with closed field lines [91] and in a cylindrical geometry with braided field lines [26]. We can also consider continuous fields rather than knotted or linked curves, e.g. knotted fluid flows and magnetic configurations [148, 149, 91, 55, 26]. Crossing number can be defined for a continuous field by averaging the crossing number of all pairs of field lines [91, 26]. This average crossing number will have some positive minimum amongst all fields with the same field line topology. Minimum crossing number and minimum field energy will then both measure the topological complexity of the field. For example, a closed, knotted tube of magnetic flux will have a magnetic energy which generally increases with the Cmin of the knot (and with internal twist of field lines inside the tube). There has recently been a major effort to find the ideal shapes of knots. While the definition of ”ideal” varies, the ideal shape is mostly obtained by minimizing some form of knot energy. Various energy functionals have been suggested for knotted curves [91, 115, 187]. These energy functionals have a positive minimum depending on the knot type analogous to minimum crossing number. Some theoretical questions arise from this work. For example, are the energy minima found using these approaches local or global minima? One would like an energy such that the minimum is global for any initial configuration. This does not seem to be possible, however [63]. Another important question is whether an energy minimum corresponds to the minimum of a more traditional measure of complexity, for example the crossing number. As mentioned above, it has been implicitly assumed in the literature that this is true, however we argue here on the basis of statistical results that this is not true. As mentioned above, energy can be defined in many ways and different energies behave differently. We consider energy to be the length of the strings in the braid. Braids, Knots and Applications 85 We look for a force that will minimize the crossing number of a braid. It can be shown [23] that this force is Fi (z) = −λǫ2  [(2r′ r′ − rr′′ ) · (ẑ × r)] (rẑ × r) )3/2 ( 2 j=i (r′ · ẑ × r) + r4 ǫ2 (142) √ where r = rij (z), r = r · r, r′ = (r′ · r)/r and ẑ is the unit vector in the z direction. What we observe in practice is that this force causes the strings of the braids to move apart from each other and prevents equilibrium from being reached. Thus we apply the additional constraint that  xi (z) · xi (z) ≤ R (143) i where R is a parameter of the model. After imposing this we can agree to have reached equilibrium if and only if the maximum distance moved by a point on the braid at any time step is less than another parameter η. Two energy minimizing approaches were tested with respect to minimizing crossing number. Both minimize elastic energy but they differ essentially in the way the elastic force is implemented: a nearest neighbor approximation (the constrained elastic force) versus a tension force depending on the curvature (the curvature elastic force). As these forces treat the strings as elastic, they pull the strings closer together and would cause them to intersect and thereby change topology. In order to prevent this, we shall introduce a repul(r) (e) sive force Fi (z, t) to the elastic force Fi (z, t) to make up the total force which we use to simulate the braids, (e) (r) Fi (z, t) = Fi (z, t) + Fi (z, t). (144) For the purposes of the repulsive force, we imagine the strings to be of circular cross-section with diameter d. We define this repulsive force by .  0 for |xi − xk | > d (r) Fi = (145) xi − xk (d − |x − x |) otherwise. i k |xi − xk | k=i Since the repulsive force is non-zero in only a limited number of cases, computing it is relatively fast as opposed to using a potential function. If we imagine the points of the geometric braid to be beads of mass m connected by springs of spring constant k and zero natural length, the elastic force on the j th bead due to the two springs attached to it is (considering only nearest neighbor interactions)          j j j+1 j−1 (e) , t = −k 2xi , t − xi , t − xi ,t . (146) Fi bc bc bc bc 86 Patrick D. Bangert This is the constrained elastic force. As given in equation 146 the constrained elastic force is a finite difference scheme for the differential equation (e) Fi (z, t) = k d2 xi (z, t) . dz 2 b2 c2 (147) Once the total force is known, we apply it to the beads xi (z, t + δt ) = xi (z, t) − Fi (z, t) 2 (δt ) . 2m (148) We have neglected the fact that beads should acquire a velocity after the force is first applied. Ignoring this velocity serves to heavily damp the system, which is desirable for the simulation. A proof that this is acceptable on a fundamental level is given in [24] and references therein. The force is applied for a duration of δt after which the beads will have moved a certain distance. The maximum distance moved by any bead in the whole braid during any step r(t) decreases monotonically to zero since the system is heavily damped due to the neglection of the velocity and that fact that the springs have natural length zero. If no bead moves more than a minimum distance of η, we may terminate the simulation because in all subsequent steps of the simulation no bead will move further than η. Thus the end of the simulation is reached when r(t) ≤ η. A given braid will determine n and c but we have endowed the model with a number of parameters: The string diameter d, the number of beads per crossing b, the mass of a bead m, the spring constant k, the separation of the strings δx , the duration of the force δt and the equilibrium distance η. Based on computer experiments we make choices for some of these d= δx , 6 0.1 ≤ k < 0.5, δt2 = 2m, 10 ≤ b ≤ 50, δx , 105 1 . δx = n−1 η= (149) (150) These values have been found to give good results. With increasing k, fewer steps are required to reach equilibrium but k < 0.5 must be observed because otherwise the repulsive force will not be very successful. Accuracy increases with b but so does the computation time. Setting b < 10 will fail because the distances between beads are large enough for the repulsive force not to guarantee isotopy, however b > 50 is unnecessarily expensive in terms of time. The other way of dealing with elastic relaxation is to treat each string in the braid as a bungee cord, subject to a tension force which aims to reduce any curvature and bring back the string to a straight configuration (given the constraint on the end points). Indeed, as already remarked above, a repulsive force among the strings is needed to counteract the tension in order to maintain the topology unchanged. We obtain a new elastic force from a Lagrangian action minimizing the length of the braid [23]. The result is the horizontal force Braids, Knots and Applications 87 d 2 xi d2 z dxi . (151) − 2 2 ds ds dz This is the curvature elastic force. This force moves the curve as the full curvature force would; the second term gives an extra horizontal displacement to the string which compensates for the effect of the missing vertical force. Once its value is known in each point of the braid and cumulated with the repulsive term, advancing in time is achieved according to the same scheme as above (148). The actual evaluation of the curvature elastic force involves the computation of second and first order spatial derivatives. In this case, then, we found it convenient to use a grid of N evenly spaced points along the z axis and ordinary centered difference. Stopping criteria for the numerical simulation of energy relaxation were defined as explained in the previous section. We kept N = 200 in all the cases presented in the following, while the choice of the other parameters was (e) Fi = d = δx , δx η = 5, 10 δt2 = 2m, δx = 1 20 . (152) (153) Extensive numerical calculations were made in order to compare the above methods of finding the minimum crossing number of a braid. A large selection of random braidsirandom braid were generated as discussed in §2 and then simulated using all four different methods described above. We have investigated, by means of computer simulation, different methods to reduce the crossing number of a braid over its equivalence class. As a group theoretical question, this problem is difficult (if the minimum is to be found [168]) but can be profitably approached using a heuristic algorithm presented above. A braid can also be regarded as a topological object divested of this algebraic approach. Here the strings may move (except the endpoints) in the embedding manifold without crossing each other. For algorithmic purposes a systematic way to move the strings must be found based on certain principles. Two of our approaches center on a physical model of the strings as elastic strings made of flexible material. Elasticity may be modelled using a nearest neighbor or curvature approach, both of which were investigated. Another way to systematically move the strings is to construct a force not based on a physical idea but by using the crossing numbericrossing number (as an integral) as a potential in a LagrangianiLagrangian. This last approach has proved to be the most successful in terms of finding the shortest braid, on average. It is however the most time consuming method. The algebraic approachialgebraic algorithm, while only third (out of the four methods) in reduction efficacy, is the fastest by far. In many applications, the braid is already an embedded topological object and not an element of the braid group. Here the two energy methods find their application as they are the only physically relevant methods. In solar 88 Patrick D. Bangert physicsisolar physics, for example, the magnetic field lines may be modelled as braids. These seem to behave as elastic configurations over time. It must be mentioned that the endpoints of these braids do move but in a random fashionirandom braid. Research about this added complication is in progress. In physical applications, we are most concerned about the energy of a braided configuration and the elastic model seems to be the most realistic for a variety of applications. While the constrained approach is more successful in terms of crossing number, the curvature fares better in an energetic sense. What has clearly emerged from the discussion above is that minimum energy and minimum crossing number for braids are different things. While reducing energy does also reduce crossing number, reducing crossing number does not necessarily reduce energy and crossing number may be reduced much further after the minimum energy configuration has been reached. Thus, it is clear that the elastic approaches terminate in a local minimum as far as the equivalence class of the initial braid is concerned. From the point of view of ideal knot theory, this result is significant because it has often been suggested that by reducing some form of knot energy, one may find a knot which is particularly simple over its equivalence class. Whether this measure of simplicity coincides with minimum crossing number over all possible projections (the traditional measure of simplicity used by knot tabulators such as Tait) has given rise to some debate for which our result provides additional fuel. We conclude our investigation by saying that the algebraicialgebraic algorithm method provides a useful minimization approach for purely group theoretical work, the crossing force is the best approach when one wishes to find an especially short braid (and is not bound to a purely group theoretical framework) and the curvature elastic energy is the best scheme to minimize elastic braid energy, i.e. total string length. 5.4 Feynman Path Integrals In classical physics, one writes down the Lagrangian of the system which is the kinetic energy minus the potential energy of all the parts of the system. The action S is the integral over time of the Lagrangian and one uses the “principle of least action” to say that nature will acts so as to minimize the action. By the calculus of variations, one arrives at the Euler-Lagrange equations in terms of the kinetic and potential energy expressions and these are the equations of motion of the system. When we integrate over time from some initial time t1 to some final time t2 , the initial conditions together with the equations of motion prescribe one definite path of all particles from their initial to their final states. In quantum mechanics however, the particles do not travel along a welldefined path as their positions and momenta are not simultaneously absolutely certain (according to the von Neumann and Copenhagen interpretations). Usually, this illustrates that the concept of a path is not useful in quantum mechanics. It was through a hint of Dirac that Feynman had the Braids, Knots and Applications 89 idea of turning this limitation around. If we are not certain of the path, we should integrate over all possible paths [76, 77, 87, 88, 89]. However not all paths are equally likely. Feynman’s idea is that each path should obtain a weight of exp(iS/
) where S is the action for that path and
is Planck’s constant divided by 2π. We recall that a quantum mechanical probability is obtained by taking the absolute value squared of the wave-function. How then is this weighting of any use as | exp(iS/
)|2 = 1? The answer is that all these paths must be added “coherently,” i.e. inside the absolute value sign. The paths act like waves which interfere with each other and there will be constructive and destructive interference in various places. The classical path is that path which extremizes the action and so there will be heavy constructive interference in its neighborhood. In this way, the classical path is reproduced in the classical limit. From a strict mathematical point of view, the path integral to be done is not well-defined as no measure theory exists which makes it precise [123]. Nevertheless it can be evaluated and is of particular use in numerical work as it does not involve operators but merely classical functions. In fact, the search for a formulation of quantum mechanics amenable to numerical calculation leads to this approach [6]. Because of the difficulties involved, many mathematicians study the path integral from a formal basis with a view to make it rigorous [95, 162] and have had some success in the semi-classical expansion of the integral in powers of
[152, 153, 154, 155, 69, 70]. For more background and comprehensive introduction to path integration see [123]. Path integrals are very useful in polymer physics. Several polymers may entangle giving rise to links or self-entangle giving rise to knots. One may make a computer simulation of polymers from this approach and discover that the probability p of a self-entangled polymer being topologically equivalent to the unknot or unlink is p ≈ 1.2325 × 0.9949N (154) where N is the number of segments of the polymer (for example base pairs in DNA) [145, 146, 207, 212, 126]. In other words, the probability of an unlinked self-entangled polymer decreases exponentially with the length of the polymer. Via path integration we can approach DNA recombination (from the Calugareanu theorem), the Chern-Simons theory which gave rise the Witten invariants of topology and anyons (including the fractional quantum Hall effect) [123], all of which will be discussed in the following sections. Path integrals provide a natural way to use topological invariants defined as integrals in physical systems described by a path integral. It is conceivable that much new topology and physics could be derived from combining the Feynman approach with the integral formulation of knot theoretic invariants by Kontsevich [127]. 90 Patrick D. Bangert 5.5 Braids and Particle Physics: Anyons It is the common wisdom of the Copenhagen interpretation of quantum mechanics that the world’s building blocks act sometimes as waves and sometimes as particles. For lack of a better word, these entities are usually referred to as “particles” and they come in two varieties: bosons and fermions. Any particle or aggregate of particles is described by a complex valued function. What matters physically is the absolute value of this function and so multiplying by a factor, known as a “phase,” of exp iθ for some phase angle θ is immaterial in the sense of causing no different physical behavior of the system. Given a collection of identical particles, swapping any two of them should give the identical physics. The only thing which may conceivably change is the phase of the function. We call particles bosons if exp iθ = 1 and fermions if exp iθ = −1. This distinction is very fundamental and gives rise to a host of other differences which are felt physically. For example, only fermions obey an exclusion principle in which two identical particles may not be in the same state. This gives rise to the electronic structure of atoms and thus chemistry. Curiously, all particles which may be thought of as matter are fermions; such as electrons and quarks (note that we are talking of fundamental particles so that protons and neutrons do not figure). Correspondingly all particles which are thought of as force carriers, the result of the quantization of the field, are bosons; such as the photon, gluon, W and Z and the mysterious graviton. The mathematical reason for this distinction and that there are only two possibilities is the following. As the particles are identical, we can swap them by merely relabelling them. Thus we have a representation of the symmetric group on the space of all possible configurations such that it maps any permutation to a scalar multiple of the identity. There are clearly only two such representations: The trivial one which does nothing and the one which switches sign (bosons and fermions respectively). All particles observed in nature appear to be of one type of the other. This is rather curious as there is no fundamental reason to request a representation of the symmetric group. It seems natural based on some sort of Diracian beauty principle but it is restrictive. Only a projective representation is required by quantum mechanics and in this case we get the full spectrum of phases. The particles for which exp iθ = ±1 are referred to as “anyons.” If they exist, why are they not observed? In relativistic quantum field theory we may prove a so called spin-statistics theorem which shows that anyonic behavior can be arrived at by combining fermions and bosons so that there really are no anyons. This acts as Occam’s razor until we note that the theorem only holds for space-times of dimension greater than three. As we live in one of these, we are done. Except that space-times of smaller dimensionality may be simulated in the laboratory; for example in a very thin slab of some material which is then a three dimensional space-time (two space, one time dimension). Braids, Knots and Applications 91 An experimental observation known as the fractional quantum Hall effect may prove to be the vindication of the anyonic hypothesis. The Hall effect is the observation that a magnetic field applied perpendicularly to a current in a wire forces the electrons to one side. One sees this from the electric field which, for a good conductor, increases like a step function and hence is quantized. The catch is that the size of the steps is sometimes less than what one expects and this is called the fractional quantum Hall effect. The most convincing theory of this effect which explains the observed data makes integral use of anyons. The details of all this are beyond the scope of this section. What remains is to make clear what all this has to do with braids. Before, we merely switched the labels of the particles (as they were identical) and got a different state function. Imagine actually physically moving two particles and switching their positions. Again, we can arrange the phase difference to be anything and thus get anyons. If we draw world-lines of the switching process, we get a braid. Thus we have exchanged the symmetric group representation which gave us bosons and fermions with a braid group representation which gives us anyons (which contain bosons and fermions as a special case). Anyons can be constructed in several ways. When a system includes many particles (a gas for example) the statistical behavior of the particles becomes important. The concept of particle statistics can be visualized by an example from the history of war. At the battle of Waterloo the English fought against the French. The French generals lined their troops up in several rows. The reason was that when the first row had shot, they would bend down while reloading and the second row could shoot. In this way, the French could fire more rapidly than the English who lined their troops up in one long row. The English fired once and obliterated the enemy. On average, the number of shots per minute could be the same but shooting all at once at the beginning means that many of the next rows of the French troops could no longer shoot and were thus defeated. The English act like bosons and the French like fermions in this example. Bosons are allowed to be in the same state as other bosons whereas fermions have to have at least one different quantum number (Pauli exclusion principle) and are thus schematically arranged in several rows. One would like to know if there is some method to interpolate such behavior. Anyons can have “any” statistics between bosons and fermions. The statistics arises from a symmetry of the wave-function describing the particles; symmetric bosons and anti-symmetric fermions. One first wants to get a clear understanding of what is meant by the interchange of two particles. By means of the path integral formulation of quantum theory, an interchange is thought of as a real physical moving of the two particles [128, 68]. It is found that the phase factors gained from different interchange paths generate the fundamental group of the configuration space of the system [161]. The fundamental group of the configuration space (R2(N −1) /SN with N particles and SN the symmetric group of N particles.) of identical particles in 92 Patrick D. Bangert the plane is the braid group [86, 90]. The Feynman path integral one must compute for this sort of theory is over all closed paths in the configuration space, in other words, the closures of the braid paths. Close connections to braid and knot theory can be established on this basis. One can also consider the problem in more complex spaces (see the references in [160]). Another route to anyonic behavior is via Schrödinger quantization of many-valued wave-functions in dimension two or in mathematical language vector bundles [205, 75, 161, 189, 137, 216, 217, 133, 22]. Further weight is given to this by noting that in a multiply connected space, a complex valued wave-function may become multi-valued [160]. It is remarkable that one can meaningfully do physics in one or two dimensions seeing as we are living in a world of obviously more dimensions. The reason that this is possible is that on the one hand, it is possible to manufacture very thin (virtually one or two dimensional) materials and on the other hand because of the second law of thermodynamics degrees of freedom are frozen out as the temperature of a system approaches absolute zero. If one makes a connection between observations of fractionally charge particles and fractional statistics (still under debate), then one may say that anyons have been directly observed in the fractional quantum Hall effect mentioned earlier [56, 182, 53, 119, 99, 177, 71, 120]. The theory of the fractional quantum Hall effect is fascinating and is a substantial field of research [102, 14, 131, 101, 121, 122]. An interesting connection is that anyons can be thought of as carrying magnetic flux as well as electric charge and then the interchange phase would give rise to an Aharonov-Bohm effect [209, 210, 80, 4, 5]. This lead to the name “anyon.” One of the concrete prediction of distinct behavior of anyons to other particles is that their ground state energy of the multiple anyon harmonic oscillator increases as √ N 2 with the number of anyons N whereas the fermion energy increases as N N [160]. The ideal gas law states that P V = RT where P is the pressure, V the volume, R the molar gas constant (8.314 JK −1 mol−1 ) and T the temperature of the gas. This being for ideal gases, a real gas has P V = RT + ∞  An+1 P n (155) n=1 where the An are known as the virial coefficients (A2 is known as the second virial coefficient and so on) and the equation 155 as the virial expansion (called so because of its relation to the virial theorem [203]). Computing the virial expansion for the first few terms is a serious challenge for the anyon gas. Only the second virial coefficient can be obtained exactly in some cases and all others must be calculated from a Monte Carlo approach. See [160] a references therein for a good review of this effort. A curious implication is Braids, Knots and Applications 93 that a gas of charged anyons will develop spontaneous magnetization also in the absence of an external magnetic field [160]. 5.6 Statistical Mechanics: The Yang-Baxter Equation Mechanics is study of motion of individual objects, usually under the influence of a central conservative force such as gravity. As we know, the two-body problem can be analytically solved but already the three-body problem raises insurmountable difficulties for exact solution. If we wish to study a gas, we need to take into account a great number of individual atoms and molecules making an exact analysis impossible. The idea of statistical mechanics is that while the description of the individual particles may be difficult, describing the global properties of their collective may not be so hard. After all, it is these global properties such as temperature and pressure that physics is interested in knowing. In short, the modelling paradigm is to sacrifice the details in favor of the large picture. Statistical mechanics is a huge topic which intersects many practical concerns of technology. We shall study a small sliver of this subject here to illustrate one of its encounters with topology via the theory of braids. A few physical concepts will have to be reviewed for this to develop its full impact. According to quantum theory, an atom is allowed to exist in one of many discrete energy levels. Suppose that our gas has Ni atoms in energy level Ei for 1 ≤ i ≤ m. We wish to know the distribution of these energies so that we may calculate the energy of the whole gas. According to the MaxwellBoltzmann distribution law (which lies at the foundations of the subject and is experimentally verified to a high degree of accuracy), this is
Ei  gi exp − kT Ni = ' (156) m
E  N i gi exp − kT i=1 where N is the total number of atoms present, k = 1.3807 × 10−23 is Boltzmann’s constant (in units of Joules per Kelvin), T is the temperature of the whole gas and gi is the statistical weight of the energy level Ei (i.e. the number of distinct quantum mechanical states of a atom of the gas with the same energy). The denominator as a whole is known as the partition function, Z, m   Ei gi exp − Z= kT i=1  (157) If we have a large number of such gasses, then the average energy of such a gas can be determined to be E = kT 2 ∂ ln Z ∂T (158) 94 Patrick D. Bangert Likewise, the other global properties can be obtained directly from the partition function. Knowledge of the partition function thus gives knowledge of the global properties of the system at hand. It is thus the goal of statistical mechanics to aid us in finding Z for any system we might encounter. Naturally, for any system actually encountered in nature, this is impossible in an exact sense. There exist, however, several simple models of systems for which an exact evaluation of Z is possible. Such models are known as exactly solvable models. A classical example of such a situation is in the model of real gasses in which one obtains the van der Waals equation of state from the partition function. More information about statistical physics is available in the literature [140]. The connection with topology comes through this partition function. We will learn that the topology of the interaction of two atoms in particular models yields an invariant closely related to the Jones polynomial (in fact a generalization of the Jones polynomial). Imagine a point interaction. Two particles a and b impact at a point, interact in some way and depart, possibly in a changed state, as particles c and d. For our purposes here, let a, b, c and d be the spins of the particles involved. In general, our system allows for many spin values and so the transition ab amplitude of this interaction is a tensor indexed by the four spin values, Rcd where we have made the convention that the “inputs” are top indices and “outputs” bottom indices. If we have several interaction vertices, then we must multiply the amplitudes and sum over all possible spin values of the internal vertices. To complete the picture, we need to decide what to do with a line segment (i.e. evolution without interaction). Suppose the two ends are particles a and b, we represent this by δba which the Kronecker delta (valued at 1 if a = b and 0 if a = b). Change without interactions is not allowed (thus giving rise to the zero) and no change is represented by the one. Now we are ready to code braid diagrams in this way. ab where a and b are the inputs, For a positive crossing (like σ)i) write Rcd i.e. the bottom two line segments, and c and d the outputs or the top two ab line segments. For a negative crossing write Rcd . In a braid group, we have two relations that these amplitudes need to satisfy. We simply rewrite them σi σj ≈ σj σi (|i − j| > 1) σi σi+1 σi ≈ σi+1 σi σi+1 ⇒ ⇒ ef ab Ref Rcd = δca δdb ab lc km Rkl Rmf Rde = (159) bc ak ml Rkl Rdm Ref (160) where we have made use of the usual Einstein summation convention (if the same index appears once in the top and once in the bottom slot, sum over all values for that index). Equation 159 means that RR = I, i.e. that R is the inverse of R. Thus the inverse crossing gets the inverse scattering amplitude. Lastly we add the physical principle of spin conservation, which says that the ab we further have total spin in any interaction is conserved so that for any Rcd that a + b = c + d. Braids, Knots and Applications 95 Equation 160 is called the Yang-Baxter equation in statistical mechanics. It seems that since the Yang-Baxter equation is simply a translation of topological relations that solutions to it will yield topological invariants. This is indeed what we shall find. It is curious that in statistical mechanics the Yang-Baxter equation is a sufficient condition for a model to be exactly solvable. This immediately gives us a connection between topological invariants and exactly solvable models for physical systems. Before we go on, here is a slightly different way of constructing the connection: We note without proof that adding the relations σi2 = e will transform Bn into Sn , the symmetric group on n elements. Thus the braid group is the most natural generalization of permutations in which permuting makes real changes (cf. anyons). Now consider a vector space V and a representation of Bn on V n , the nth tensor product of V and consider the linear maps ηi : V n → V n defined by v1 ⊗ v2 ⊗ · · · ⊗ vn → v1 ⊗ · · · T (vi ⊗ vi+1 ) ⊗ · · · vn (161) where T is an invertible linear transformation of V 2 . We get the representation we want by now mapping σi → ηi . Naturally this only works if (T ⊗ I)(I ⊗ T )(T ⊗ I) = (I ⊗ T )(T ⊗ I)(I ⊗ T ) (162) where I is the identity on V . This equation is known as the Yang-Baxter equation. Recall that one of braid group relations was the bridge relation, σi σi+1 σi = σi+1 σi σi+1 (163) The similarity of the Yang-Baxter equation and the bridge relation is striking especially as we constructed the Yang-Baxter equation from braids. It was first discovered by R. J. Baxter in the context of exactly solved models in statistical mechanics [25]. Yang arrived at it by trying to find twodimensional quantum field theories for which the scattering matrix could be exactly calculated. The equation thus provides a fundamental requirement for exact solubility in a variety of areas of physics and a significant effort was started to find solutions to it. Now we shall construct the partition function for a closed braid. Recall first of all that any knot can be transformed into a closed braid and furthermore a closed braid with upward orientation (if the knot was oriented which we shall assume here). Given the braid β ∈ Bn we label all the arcs coming in and going out of any crossing and associate to each a transition tensor. We also need to take care of the boundary conditions due to closure and we need to multiply be a factor to take it into account. The partition function is then,  ab Z= R±1 cd t−(a1 +a2 +···+an ) (164) S C where the sum is over all the spin states S contained in the model, the product is over all crossings C and the power of t is the boundary condition. 96 Patrick D. Bangert The variables (a1 , a2 , cdots, an ) are the state variables gives to the top parts of the closure strings of the braid β. If we work with single fermions, we have the possibility that spin is ±1 (in units of
/2) and this simple situation will allow us to work out a partition function. We shall let the transition amplitude be   1 0 0√ 0 0 0 − t 0  √ R= (165) 0 − t 1 − t 0 0 0 0 1 Exercise 5.4. Calculate the partition function for β = σ1 ∈ B2 . The solution follows. −1,−1 −1,1 1,1 There are only three contributing states: R−1,−1 , R−1,1 and R1,1 . Finally −1 doing the sums and products we obtain Z = 1 + t . The partition function is not an invariant yet but one can show that Jβ (t) = t exp(β)+1 2 1+t Zβ (166) is the Jones polynomial [158]. Note that exp(β) is the exponent sum of the braid β. By changing the model slightly, we can achieve a generalization of the Jones polynomials also [200]. What this shows is that knots and braids can be viewed as interaction models in statistical mechanics and that the conditions that guarantee and solution for the physical case also guarantee topological invariance. It becomes apparent that topologically invariant processes are, in some sense, the simplest processes available in nature. This has given rise to a plethora of research into the connections of statistical mechanics and knot invariants. The construction that we have outlined here can generalized by the use of quantum groups as was first done in [79] and extended in [195]. We may get the original Jones polynomials [111] as outlined but also the two-variable generalizations (for example the Homfly [93] and Kauffman polynomials [116]). A number of other invariants can be obtained from the Yang-Baxter equation [7, 8, 9, 10, 64]. The original construction of the connections between topological invariants and statistical mechanics was done via quantum groups [79] (for background on quantum groups see [141]). There are a number of further connections between the two subjects and that of Lie algebras which are investigated in [20, 25, 85, 109, 125]. The Ising and Potts models in statistical mechanics have been shown to yield a knot invariant known as the Arf invariant [112]. A good review of the connections between link invariants and statistical mechanics is [195]. We illustrate a further exiting connection by means of an algebra. The Temperley-Lieb algebra first appeared in 1971 in connection with the Potts model in statistical mechanics and again when the Jones polynomial Braids, Knots and Applications 97 was invented [111]. The algebra Tn is generated by ti for 1 ≤ i < n and contains the relations ti ti±1 ti = ti (167) t2i = uti ti tj = tj ti ; |i − j| > 1 (168) (169) where u is a constant. If we can set up a topological interpretation of this algebra, we would have another connection between topology and statistical physics. Let us first construct a purely formal connection. Consider the al  gebra as a module over the ring Z a, a−1 and set u = −a2 − a−2 . Further consider the mapping ρ : Bn → Tn defined by ρ (σi ) = a + a−1 ti
−1  = a−1 + ati ρ σi (170) (171) (172) The Temperley-Lieb algebra will be a representation of the braid group if the mapping satisfies the braid group relations, 
(173) ρ (σi ) ρ σi−1 = 1 ρ (σi σi+1 σi ) = ρ (σi+1 σi σi+1 ) ρ (σi σj ) = ρ (σj σi ) |i − j| > 1 (174) (175) Exercise 5.5. Verify that the relations 173, 174 and 175 hold. You will have proven that the Temperley-Lieb algebra is a representation of the braid group. For the topological interpretation, take ti to be in the shape of the elementary tangle 0 (see figure 34) with the bottom left string being string i. Just in the same way, we built braids as stacks of crossings by writing them as words in σi , we can now write words of ti and thus obtain stacks of zero tangles. Naturally t2i is the same as ti plus an unknot. This gives us the interpretation of u, it denotes an unknot. The other relations of the algebra have clear topological correspondence. Exercise 5.6. Draw all three relations of the Temperley-Lieb algebra with the above interpretation in mind. Observe that they hold topologically. Knot polynomials may be calculated using this algebra and what is known as the state model of knot polynomials [116]. Much is known about this algebra and its connections with both knot theory and statistical physics and we shall refer the reader to the literature for further details [135]. 5.7 Anyons to TQFT’s via the Hopf Invariant Consider a two-dimensional magnet. Each atom has a spin which we consider to be a unit vector, a point on the sphere S 2 . The distribution of spins will 98 Patrick D. Bangert give us some measure of the state of the magnet. By adding a point at infinity, our model is thus a map from S 2 (position of the atom in the magnet plus the point at infinity) to S 2 (spin vector of the atom). Of course this model is simplistic beyond belief but it will get the point across. Now the whole point of topology is to observe that not all maps with identical source and target spaces are created equal. The differences between them are enumerated under the heading of “homotopy.” Imagine a donut-like structure with n holes. Given two of them with n and m holes respectively, they are homotopic if and only if n = m. A sphere may be mapped to itself by wrapping it over itself, a kind of turning inside out. These induced twists are called topological solitons and the number of times the sphere is wrapped is counted by the soliton number or winding number. As always in things quantum we can temporarily create a pair of anti-particles and then annihilate them. So let there be a soliton-antisoliton pair. We will draw their short lives as a three dimensional history and we will get a knot. Most notably, we will get the Hopf link in two possible varieties. The Hopf link is the canonical closure of the braid σ12 or σ1−2 . The two cases can be distinguished by the Hopf invariant which is also known as the linking number. From the knot diagram, we see that the first variety (σi in a braid representation) has a linking number of 1 and the second (σi−1 in a braid representation) -1 (the sign is a convention and thus subject to redefinition). The linking number of two knot components is equal to one-half of the sum of these crossing contributions. In this way, the Hopf link has linking number 1 or -1, depending on the orientations chosen. The Hopf invariant may be calculated in more esoteric ways which will illustrate a few surprising connections. Gauss wrote down an integral which gives the linking number of two curves l1 and l2 . - |(x1 − x2 ) · dx1 × dx2 | 1 (176) lk (l1 , l2 ) = 3 4π |x1 − x2 | l1 l2 We will find that we may rewrite this in another form using differential geometry. This will eventually lead us to writing another integral which will yield other knot invariant, in particular the Jones polynomial. However, before we can continue, we must explain some concepts from differential geometry. For a general introduction to the subject see [100]. Imagine that you have two manifolds M and N and you have a function φ : M → N from M to N . Given some real valued function f on the manifold N , f : N → R, we are asked to construct a real valued function g on M , g : M → R. This is most easily done by composing the function φ with f , g = φ⋆ f = f ◦ φ. (177) We then call g the pullback of f by φ. Having gotten pullbacks, we need forms. Recall the gradient ∇f of a function f defined on Rn . If n = 3, we have Braids, Knots and Applications ∇f (x, y, z) = ∂f (x, y, z) ∂f (x, y, z) ∂f (x, y, z) î + ĵ + k̂ ∂x ∂y ∂z 99 (178) We would like to generalize this concept to an arbitrary manifold M . When we have done this, we will denote it by df and call it a differential 1-form. To evaluate the directional derivative in a direction v we must take the dot product ∇f · v and this necessarily invokes the metric of the space, i.e. the function that measures distance between points. We will find it more convenient to work without a particular coordinate system (and without a particular metric) and so we would like to be able to formulate this df such that it is independent of the metric, it can therefore not be a vector field. Note that the directional derivative takes in a vector field v and returns a function, the directional derivative of f in the direction v. What we are really after is a map that accomplishes this. The essential property of this map is that it is linear, i.e. ∇f · (v + w) = ∇f · v + ∇f · w (179) So we define a 1-form to be a map from the set of all vector fields in M to the set of all infinitely differentiable functions on M that is linear over this last set. So a 1-form ω satisfies two equations ω(v + w) = ω(v) + ω(w), ω(gv) = gω(v) (180) for two vector fields v and w and an infinitely differentiable real valued function on Rn . We can multiply two 1-forms together using a so-called wedge product which is denoted by ∧. This is essentially a glorified vector cross product which we will not explain in further detail. We construct a p-form by taking a linear combination of products of p 1-forms. Finally, recall that one can get the “volume” of an interval [a, b] in R by the integral ,b (181) l = dx a This dx is a simple example of a volume form. We are now ready to translate equation 176 into our new language. It can be rewritten in terms of the volume form on S 2 pulled back to S 3 . This closed 2-form on S 3 can be written as dA of some 1-form. Now integrating A × dA over S 2 and dividing by 4π gives the desired result, , 1 lk = A ∧ dA (182) 4π S2 For details of why this rewriting process works and is equivalent to Gauss’s original integral see [21] and for a general background on differential geometry in relation to algebraic topology see [42]. Actually the choice of A is arbitrary, 100 Patrick D. Bangert this is called “gauge freedom” in physics. After this rigmarole, we arrive at the same linking numbers as above. Coming back to the situation where a soliton-antisoliton pair was created, we could create it in two ways: A Hopf link of linking number 1 and -1. Classically, the two possibilities produce the same physics but quantum mechanically they do not as we must sum over all possible histories in the Feynman path integral approach and they are fundamentally distinct (topologically distinct) histories. To be exact, they differ in a phase. The associated phase is exp(lkiθ) where lk is the linking number. In this way, they act like anyons. This is an example of a theory in which the punch line lies in phase effects just like in the Aharonov-Bohm effect (in which a applied magnetic field causes phase effects to manifest in a quantum mechanical double slit experiment) [4]. Quantum mechanically, phase differences are not supposed to play a role because when probabilities are evaluated, phases disappear. Suppose the wave-function Ψ → Ψ eiφ , then the probability |Ψ |2 → |Ψ eiφ |2 = |Ψ |2 remains invariant. After the conceptual revolution that the Aharonov-Bohm effect cause in quantum mechanics and much philosophical discussion, the existence of these phase effects was experimentally verified [204, 180]. This shows that topologically different histories may cause different physical events and thus the study of topological invariants has a real practical impact. To go further, we remind ourselves that the modern approach to get equations in physics is to write down some function called a Lagrangian and then to construct an integral of this function over the space. This integral is called the action. We look for the Lagrangian that makes this action minimal and using the calculus of variations, this gives rise to a number of differential equations that are the state equations of our physics. Much physics can be done by investigating this action integral. A quantum field theory is a theory constructed in this way that makes the predictions observed in the laboratory. Generally this is heavily dependent on the metric of the space-time it assumes. In general relativity we have learned that we may not assume a space-time as a priori given but that space-time evolves together with the matter in that space-time. Thus we look for a quantum field theory that is largely independent of the metric of that space (i.e. an action integral in which the metric does not appear). Such theories are called topological quantum field theories. From elementary geometry we have learned that matrices can be thought of as transformations (i.e. a vector can be rotated by matrix multiplication). If a transformation leaves a system of points invariant, it is called a symmetry. For example, any rotation of the unit circle (center at the origin) leaves the circle unchanged and is thus a symmetry of the circle. Such symmetries may form a group and then a system is said to possess a symmetry group. Much modern quantum field theory begins with assuming a certain symmetry group and metric independent action integral (for background on this see [208, 206]). Braids, Knots and Applications 101 A matrix U is unitary if its adjoint U ⋆ and inverse U −1 are the same, U = U −1 . The symmetry group of all n × n unitary matrices is called U (n). If we add the restriction that the determinant is to be equal to one, then the matrix is special unitary and the symmetry group is called SU (n). A matrix is orthogonal if U T = U −1 where U T is the transpose of the matrix. If its determinant is equal to one also, then it is special orthogonal. The symmetry groups based on these matrices are known as O(n) and SO(n) respectively. The topological quantum field theory started by Chern and Simons took the group U (1) as the symmetry group under consideration. They construct the action SCS (A) as a function of the 1-form A,   , k 2 (183) SCS (A) = tr A ∧ dA + A ∧ A ∧ A 4π 3 ⋆ S where S is a compact oriented 3-manifold and k is an integer called the level (it is invariant under gauge transformations). In particular, we may consider the 1-form A as the vector potential of electrodynamics, i.e. dA = B the magnetic field. The action is important in quantum mechanics because the Feynman integral is an integral over exp(iA) where A is the action. In this way, we may obtain the partition function of our theory, , (184) Z(S) = exp (iSCS (A)) DA A where DA is a Lebesgue measure and A is a complicated space that depends on the manifold S and on the symmetry group chosen. Actually evaluating this integral is very difficult. On formal mathematical grounds, it has not yet been proven that such integrals even exist as no measure theory has been developed to cope with them. This does not worry the practical people however and they have achieved many things based on the Feynman approach. Consider that one may move along several closed curves in the space S denoted by γ1 , γ2 , · · · , γn and consider the weighted partition function , Zw (S) = W (γ1 , A) W (γ2 , A) · · · W (γn , A) exp (iSCS (A)) DA (185) A where W (γi , A) is called the Wilson loop and is equal to the trace of the holonomy around γi of the connection having vector potential A. A holonomy is a map of one vector bundle to another one that has been parallel transported along a given path. To give an adequate explanation of the concepts of holonomy and connection would go beyond the limits of this section. Suffice it to say that the Wilson loop can be calculated and is a function of the 1-form A and the loop γ1 and that it measures the self-interference of a particle. In the case of a U (1) theory, it is equal to the phase change as it moves around the loop and then gives rise to the Aharonov-Bohm effect. 102 Patrick D. Bangert Now we can play with different symmetry groups. If we choose U (1) we obtain the writhe of the knot L given by the loops γ1 , γ2 , · · · , γn . On the other hand, SU (n) gets us the Homfly polynomial which is a generalization of the Jones polynomial and finally SO(n) gives the Kauffman polynomial. It has been a thorn in the eye of knot theorists for a long time that many knot invariants (especially the polynomial invariants) have been defined through two-dimensional projections of knots and that no genuinely three-dimensional definitions of them were available. This result remedies this situation. It is however clear that if one wishes to obtain the Jones polynomial of a particular knot, one should stick with the old methods and not try to compute this integral. For the original discovery of these integrals see [213] and for a more pedagogical discussion see [21]. 5.8 Tangles and DNA We consider the notion of a tangleitangle and analyze the operations which are possible on it. We use tangles in constructing a new notation for knots based on Conway’s knot notation. This new notation has several advantages over existing notations. All the basic properties of the notation and algorithms to retrieve simple knot information are discussed. Procedures for putting a knot into our notation are also given. Finally, polynomial-time algorithms, which do not rely on topological deformation, are described which produce a plaitiplait and a closed braid which are isotopic to any knot given in our notation. First, we review the notion of tangles and investigate their classification. We shall then introduce the new notation, prove that all knots may be represented by it, give an algorithm to place a given knot into this notation and present a traversal algorithm which will calculate certain features of the knot. An algorithm is then given to obtain a plaitiplait and a braidibraid, the closures of which are a given knot in the new notation. Definition and Partition Consider the 3-balli3-ball B 3 and choose 2n points on its surface, which is the 2-sphere S 2 , and call the set of these points P . Attach n polygonal curves to the 2n points such that: (i) each curve intersects S 2 in exactly 2 points in P , which are its endpoints, (ii) exactly one curve may begin or end at any one point in P and (iii) no curve may 
intersect another. If the set of these curves is T , then we will call the set B 3 , T an n-tangle. In particular, we will focus on 2-tangles and so whenever we skip the n, it will be understood that we mean n = 2. Note that our requirement that the curves be polygonal excludes any wild tangles, where wild is to be understood in the usual knot theory sense. Two tangles are called equal if they are isotopic without moving the points in P . A tangle can be visualized readily by choosing the four points (named according to the cardinal points of the compass) Braids, Knots and Applications NW z • OO  / & • $ 103 NE  "  ^ ` \ Z Xy b  fd O O  o //  O X Z \x ^_ ` b d f o "  $  & • •  SW SE /  _ Fig. 33. The 3-ball and the four points on its surface which form the endpoints of the two polygonal curves necessary to define a tangle.   1 1 N E = 0, √ , √ , 2 2   1 1 SE = 0, √ , − √ , 2 2  1 1 0, − √ , √ 2 2   1 1 SW = 0, − √ , − √ 2 2 NW =  (186) (187) on the unit sphere, which will be our canonical B 3 , see figure 33. Even though tangles are, by definition, three dimensional objects, we will work with their projection onto the two dimensional plane as if the projection is the tangle. The fact that a projection in which there are at worst double points always exists for a tangle follows from the corresponding theorem about knots. • • 0 • • • • ∞ • • • • 1 • • • • • • −1 Fig. 34. The elementary tangles. We shall find it convenient to partition the set of all possible tangles into a few categories: elementary, integral, fractional, rational and irrational. The simplest are the elementary tangles, of which there are four. These are best introduced by displaying them in figure 34. Note that we have not drawn B 3 , it should however be understood to be present. The reason for naming them as they have been will become apparent later on. Note that the literature disagrees on which of the two tangles ±1 is to have the minus sign, this is a matter of convention and has no serious consequences (we follow the convention introduced by Conway). The other types of tangles can be most readily defined in terms of combining the elementary ones in some way. To do this, we shall define two ways 104 Patrick D. Bangert ?? ?  ?   ??  A  ?? A+B ≡ , ??  ?? A⊕B ≡  ??   ?? A B B Fig. 35. Tangle addition. of adding tangles. Following Conway, we denote a general tangle by an ”L” shaped symbol within the 3-ball and we also sketch the ends of the two curves by which tangles may be attached to one another. In this way, we define the horizontal sum + and the vertical sum ⊕ in figure 35. In what follows, we shall use a superscript to denote of which type a particular tangle t is; for example an elementary tangle t would be denoted by t(e) . An integral tangle t(i) and a fractional tangle t(f ) will be defined in terms of the elementary tangles ±1 by t(i) = 1 + 1 + · · · + 1    t factors t(f ) = 1 ⊕ 1 ⊕ · · · ⊕ 1    t factors (188) t(r) = a(i) ⊕ b(f ) + c(i) ⊕ d(f ) + · · · + z (i)    j odd (190) (189) The negative versions are, of course, the sums of −1 tangles instead of 1 tangles. A rational tangle t(r) can then be defined in terms of a sum of integral and fractional tangles. The definition of the sum differs if the number of tangles j in the sum is even or odd, this is because the definition requires an alternate sum between integral and fractional (and the two methods of addition) which always ends in an integral tangle being added. This is because the set of rational tangles may be classified if this restriction is imposed; the classification scheme is outlined in the next section. The integral tangles, including the last, may be zero and the fractional tangles may be infinite. If any component tangles are 0 or ∞ though, they may be removed from the sum and the terms immediately preceding and following the removed term may be added together to shorten the sum, while preserving isotopy. t(r) = a(f ) + b(i) ⊕ c(f ) + d(i) ⊕ · · · + z (i)    j even (191) Note that the set of elementary tangles is a subset of both the integral and fractional tangle sets which are subsets of the rational tangle set. We shall call any tangle which is not rational, irrational. Braids, Knots and Applications 105 Classification of Tangles We may denote a rational tangle by giving its integral and fractional factors in order. Thus a sequence of integers t(r) = (a1 , a2 , . . . , ai ) defines any rational tangle. Note again that the identity of the tangle factors is decided by requiring the last in the sequence to be integral. Given a rational tangle t(r) = (a1 , a2 , · · · , ai ), we may associate with it an extended rational number E(t(r) ) = α/β, where α and β are integers including zero. We say an extended rational number because this allows for 1/0
= ∞, the inclusion of which extends the rational numbers. We calculate E t(r) by the continued fractionicontinued fraction (the + signs are arithmetic additions and not tangle additions) ( ) 1 E t(r) = ai + (192) 1 ai−1 + 1 ai−2 · · · + a1 Conway [61] was able to deduce that two rational tangles are equal if and only if the associated extended rational numbers were equal, this is called Conway’s Basic Theorem. The first published proof may be found in [48] but a more intuitive proof was given by Goldman and Kauffman [98]. Thus Conway’s Basic Theorem classifies rational tangles in a simple algorithmic manner. In particular, the fractions associated with the elementary tangles are their numerical names: 0, ±1 and ∞. The fraction for an integral tangle t(i) is t(i) and for a fractional tangle t(f ) is 1/t(f ) . It is clear now why these tangles were named as they were. This concludes our review of previous work on tangles and the rest of the chapter is new work. By equation 192 is easy to calculate the fraction associated with a given rational tangle. Given a fraction, it is also possible to decompose it into appropriate factors, thereby constructing the rational tangle associated with it. Euclid’s algorithm will accomplish this. Connection to DNA DNA is a well known molecule which can be regarded to be the code for the generation of a human being or an animal. It is in the form of double helix. Two strands of molecules are connected to each other and wind about a common axis (this looks like figure 2 (b)). This double helix has ends which may or may not connect thus forming a knot. There also exist enzymes which break this knot locally, make some changes and glue the cut back again. In the laboratory it is possible to make pictures of knots and links of DNA. In particular it is possible to make pictures of a knot before and after such an enzyme has acted upon it. The goal for topology is to describe what changes are possible and thus to create a topological model for site-specific recombination, i.e. the cutting and gluing operations of such enzymes at a particular location. We recall that molecules with the same chemistry but different structure are isomers. Two DNA molecules with the same base pair assignment 106 Patrick D. Bangert are isomers but may have different linking numbers due to enzyme action, they may thus be topologically distinct and are thus called topoisomers. An enzyme may cut a single stand of DNA, pass the other strand through it and glue the cut back again. This is the enzyme topoisomerase I. The other possibility is that the enzyme cuts two strands of DNA, passes two strands through and glues the cut back again. This is the enzyme topoisomerase II. In
−1 −1 2 2 braid terms, this is making σi into σi−1 and making (σi+1 σi ) into σi+1 σi respectively. Both moves change topology locally. Empirical evidence tells us that almost all the products of these sitespecific recombinations are rational knots (if one starts with the unknot) and that the part of the DNA molecule which is acted upon by the enzyme is the (2, 2)-tangle. Let us call the tangle that is the DNA molecule outside of the recombination site S and the recombination site before the enzyme action E and after R. We observe that the whole DNA molecule is knot K1 before the event and K2 after the event. Then we have N (S + E) = K1 ; N (S + R) = K2 (193) Knowing K1 and K2 from observations, it is our task to find S, E and R. This is of course not possible using just two equations. Here, another empirical observation comes to our aid: The change an enzyme makes depends only upon the enzyme, not the DNA molecule. Thus if the enzyme make several changes, they are all the same. Thus if we have another observation of a knot K3 after a second change, we have N (S + R + R) = K3 (194) In the experiments actually carried out, the following was found N (S + E) = Unknot N (S + R) = Hopf Link N (S + R + R) = Trefoil N (S + R + R + R) = Whitehead Link N (S + R + R + R + R) = 62 (195) (196) (197) (198) (199) (200) As the knot 62 is chiral (it is not isotopic to its mirror image), we must be careful to specify it properly. Rolfsen [175] and Murasugi [158] show the mirror images of each other in their respective tables. We shall follow the original publication of the empirical result [81] in which they give the tangle (1, 1, 1, 2, 1) which is isotopic to the knot 62 as given by Rolfsen and as shown in this text in figure 8. Given this information, it is possible to find solutions for S, R and E. We shall merely give the results as the proofs are quite lengthy (see [81]). S turns out to be the tangle (−3, 0) which is, of course, the unknot under numerator Braids, Knots and Applications 107 closure. R is the tangle (1). The tangle E can not be uniquely determined using these equations, it is (2x, 3, 0) for any integer x. Important biological information can be extracted from this. For example, the Whitehead link can not be obtained using just a single recombination event. This investigation gives rise to much stimulating mathematics which we do not have the space to present. The theory of solving tangle equations (to which the above is mere special case) is not fully developed yet [82, 83]. Nothing has been done on solving tangle equations for larger than (2, 2)tangles. 5.9 Ideal Knots Recently there has been a great effort to study “ideal knots.” There is considerable disagreement on what “ideal” is to mean and of what use the endeavor is. We shall give a brief introduction to the problems involved here. For the topologist, a knot is a topological object which does not change under continuous transformations. For the geometer or physicist such a transformation is real. What one would like is a canonical representative of each topological knot type; a definite embedding for each knot type which is going to serve as an ambassador for the type. This ambassador should have the following properties: (1) There should be exactly one for each knot type, (2) there should be an (efficient) algorithmic prescription to find it for any knot type (i.e. an algorithm is started with a knot and evolves it until it has reached this canonical type) and (3) this prescription should always find this one configuration no matter with which initial embedding the algorithm begins. Note that if one is successful in finding such a prescription, one has solved the knot classification problem. Needless to say, these properties are only approximated by the currently known approaches. A further desire is that the canonical type should be (4) simple and (5) aesthetically pleasing. If the knot has symmetries, the canonical embedding should make these apparent. A particular knot embedding which may serve as such an ambassador for its type is called ideal. As no known algorithm exists that satisfies all five desiderata, one must prefer one point over another and this is where the different meanings of “ideal” have their origin. The historical start of the discussion can be well-defined: It happened in a conference talk given by Jonathan Simon in Nov. 1991 in Bielefeld in which he called a number of knot conformations produced earlier “ideal” [184]. The most common approach is to begin with a definite embedding of a knot and to evolve it using a physical model for it. This typically proceeds by first writing down an expression that will be called the energy of the knot; typically an integral over the knot of some quantity. This “energy” is to be minimized just like classical physics and so we use the calculus of variations and obtain some equations that will allow us to construct an energy reducing transformation. Having reached the minimum energy state, we ask whether it is a local minimum or the global minimum. We ask if a definite 108 Patrick D. Bangert global minimum exists. We ask if we can reach this global minimum from all points in state space (all initial embeddings) using the algorithm obtained. In general, the answer to these questions can not be fully given. Various energies have been investigated and have been observed (by numerical experiments) to have some advantages and disadvantages. A very simple way to define idealness is to look at the largest radius that the knot tube can have without overlapping with itself divided by its total length. This can obviously be formulated via a minimum principle by taking the “energy” of the knot to be its length (the knot being made of a rope with some finite thickness). This is a very interesting energy as it correlates with the speed with which knotted DNA polymers move through a gel. It is an experimental observation that knotted DNA polymers move faster than unknotted ones and the minimum rope length measure of a knot type is linearly proportional to the gel velocity of a DNA knot of that type [188]. However the gel velocity varies with the conditions under which the knots move through the gel and so the coefficient of the linear relation is a function of the conditions and the knot type [134]. After the energy approach was invented [164], many energies were tried [149, 47, 92, 185, 165] but it was found that energies are actually difficult to define given that they have to obey some basic laws such as approaching infinity when the knot approaches a self-intersection [72, 73, 74]. Other methods to obtain ideal knots also exist. Simulated annealing is a popular way to find minima numerically and investigations to determine its usefulness for knots are under way [132]. A very sophisticated knot tightening algorithm called Shrink-On-No-Overlaps (SONO) which was invented while studying braids can currently be called the best ideal knot algorithm [170]. It is too complex and technical to present here; let it just be said that the method is not elegant but the results are even though the algorithm gets stuck in local minima [169]. An experimental vindication of the SONO method was performed and confirms its results [74]. An alternative way to view idealness is by a a functional representation. It can be shown that every knot is harmonic, i.e. every knot can be represented by three functions (giving the three coordinates of its points) of an arc length parameter t which are all finite trigonometric polynomials [193, 194]. 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