The mitochondrial DNA of Trypanosomes, known as kinetoplast DNA (kDNA), is organized into several thousands of minicircles that are topologically linked, forming a large chainmail-like network. How and why minicircles form a network in... more
The mitochondrial DNA of Trypanosomes, known as kinetoplast DNA (kDNA), is organized into several thousands of minicircles that are topologically linked, forming a large chainmail-like network. How and why minicircles form a network in some but not in other kinetoplastid organisms, remain unanswered questions. Motivated by these questions, in our earlier studies we introduced some simple analytical and numerical models to study networks of topologically linked minicircles. In these earlier studies, we used three key parameters, namely the mean minicircle valence (i.e. the number of minicircles topologically linked to any given minicircle in the network), the critical percolation and mean saturation densities to characterize the topological properties of the minicircle network and how these properties change with respect to the minicircle density changes. Using these models we showed, both theoretically and numerically, that high minicircle density leads to the formation of a linked ...
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It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, we do not even know that... more
It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, we do not even know that Cr(K1#K2) = Cr(K1) or Cr(K1#K2) = Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. The best known result to date is the following: if K1 and K2 are any two alternating links, then Cr(K1#K2) = Cr(K1) + Cr(K2), A less ambitious question asks for what link families the crossing number is additive. In particular, one wonders if this conjecture holds for the well known torus knot family. In this paper, we show that there exist a wide class of links over which the crossing number is additive under the connected sum operation. We then show that the torus knot family is within that class. Consequently, we show that Cr(T1#T2# · · ·#Tm) = Cr(T1) + Cr(T2) + · · ·+ Cr(Tm). for any m ≥ 2 torus...
A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all... more
A rational link may be represented by any of the (infinitely) many link diagrams corresponding to various continued fraction expansions of the same rational number. The continued fraction expansion of the rational number in which all signs are the same is called a {\em nonalternating form} and the diagram corresponding to it is a reduced alternating link diagram, which is minimum in terms of the number of crossings in the diagram. Famous formulas exist in the literature for the braid index of a rational link by Murasugi and for its HOMFLY polynomial by Lickorish and Millet, but these rely on a special continued fraction expansion of the rational number in which all partial denominators are even (called {\em all-even form}). In this paper we present an algorithmic way to transform a continued fraction given in nonalternating form into the all-even form. Using this method we derive formulas for the braid index and the HOMFLY polynomial of a rational link in terms of its reduced altern...
Let [Formula: see text] be the set of un-oriented and rational links with crossing number [Formula: see text], a precise formula for [Formula: see text] was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration... more
Let [Formula: see text] be the set of un-oriented and rational links with crossing number [Formula: see text], a precise formula for [Formula: see text] was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] and let [Formula: see text] be the set of oriented rational links with crossing number [Formula: see text] ([Formula: see text]) and deficiency [Formula: see text]. In this paper, we derive precise formulas for [Formula: see text] and [Formula: see text] for any given [Formula: see text] and [Formula: see text] and show that [Formula: see text] where [Formula: see text] is the [Formula: see text]th convolution of the convolved Fibonacci sequences.
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It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of... more
It is well known that the minimum crossing number of an alternating link equals the number of crossings in any reduced alternating link diagram of the link. This remarkable result is an application of the Jones polynomial. In the case of the braid index of an alternating link, Yamada showed that the minimum number of Seifert circles over all regular projections of a link equals the braid index. Thus one may conjecture that the number of Seifert circles in a reduced alternating diagram of the link equals the braid index of the link, but this turns out to be false. In this paper we prove the next best thing that one could hope for: we characterise exactly those alternating links for which their braid indices equal the numbers of Seifert circles in their corresponding reduced alternating link diagrams. More specifically, we prove that if D is a reduced alternating link diagram of an alternating link L, then b(L), the braid index of L, equals the number of Seifert circles in D if and on...
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ABSTRACT This article reports the results of an investigation into the average behavior of the knot spectrum (of knots up to 16 crossings) of a family of random knot spaces. A knot space in this family consists of random polygons of a... more
ABSTRACT This article reports the results of an investigation into the average behavior of the knot spectrum (of knots up to 16 crossings) of a family of random knot spaces. A knot space in this family consists of random polygons of a given length in a spherical confinement of a given radius. The knot spectrum is the distribution of all knot types within a random knot space and is based on the probabilities that a randomly (and uniformly) chosen polygon from this knot space forms different knot types. We show that the relative spectrum of knots, when divided into groups by their crossing number, remains unexpectedly robust as these knot spaces vary. The relative spectrum for a given crossing number c is Pc(u)/Pc, where Pc is the probability that a uniformly chosen random polygon has crossing number c, and Pc(u) is the probability that the chosen polygon has crossing number c and is from the group of knots defined by the characteristic u (such as “alternating prime,” “nonalternating prime,” or “composite”). Specifically, for a fixed crossing number c, the results show that tighter confinement conditions favor alternating prime knots, that is Pc(A)/Pc (where A stands for “alternating prime”) increases as the confinement radius decreases, and that the average Pc(N) (where N stands for “nonalternating prime”) behaves similar to the average Pc − 1(A). We then use our simulations to speculate on limiting behavior.
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In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these... more
In this paper we define a set of radii called thickness for simple closed curves denoted by K, which are assumed to be differentiable. These radii capture a balanced view between the geometric and the topological properties of these curves. One can think of these radii as representing the thickness of a rope in space and of K as the core of the rope. Great care is taken to define our radii in order to gain freedom from small pieces with large curvature in the curve. Intuitively, this means that we tend to allow the surface of the ropes that represent the knots to deform into a non smooth surface. But as long as the radius of the rope is less than the thickness so defined, the surface of the rope will remain a two manifold and the rope (as a solid torus) can be deformed onto K via strong deformation retract. In this paper we explore basic properties of these thicknesses and discuss the relationship amongst them.
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... C. Rousseau Multisection Revisited Robert Rubalcaba Efficient Cartesian Product Layer Domination Attila Sali Partition critical hypergraphs Richard H. Schelp Some Ramsey Results and Conjectures Akos Seress Triangle-avoidance games... more
... C. Rousseau Multisection Revisited Robert Rubalcaba Efficient Cartesian Product Layer Domination Attila Sali Partition critical hypergraphs Richard H. Schelp Some Ramsey Results and Conjectures Akos Seress Triangle-avoidance games James Shook A characterization of ...
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Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly... more
Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots.
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Let a, b be two fixed non-zero constants. A measurable set E ⊂ R is called a Weyl-Heisenberg frame set for (a, b) if the function g = χE generates a Weyl-Heisenberg frame for L(R) under modulates by b and translates by a, i.e., {eg(t −... more
Let a, b be two fixed non-zero constants. A measurable set E ⊂ R is called a Weyl-Heisenberg frame set for (a, b) if the function g = χE generates a Weyl-Heisenberg frame for L(R) under modulates by b and translates by a, i.e., {eg(t − na}m,n∈Z is a frame for L (R). It is an open question on how to characterize all frame sets for a given pair (a, b) in general. In the case that a = 2π and b = 1, a result due to Casazza and Kalton shows that the condition that the set F = ⋃k j=1 ([0, 2π) + 2njπ) (where {n1 < n2 < · · · < nk} are integers) is a Weyl-Heisenberg frame set for (2π, 1) is equivalent to the condition that the polynomial f(z) = ∑k j=1 zj does not have any unit roots in the complex plane. In this paper, we show that this result can be generalized to a class of more general measurable sets (called basic support sets) and to set theoretical functions and continuous functions defined on such sets.
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Abstract In this paper we study the average crossing number and writhe of random freely-jointed polygons in spherical confinement. Specifically, we use numerical studies to investigate how these geometric quantities are affected by... more
Abstract In this paper we study the average crossing number and writhe of random freely-jointed polygons in spherical confinement. Specifically, we use numerical studies to investigate how these geometric quantities are affected by confinement and by knot complexity within random polygons. We report and compare our results with previously published results on knotted random polygons that are unconfined. While some of the results fall in line with what have been observed in studies of unconfined random polygons, some surprising results have emerged from our study, showing properties that are unique due to the effect of confinement. For example, under tight confinement, the average crossing number and the squared writhe grow proportional to the polygon length squared. However, the squared writhe of polygons with a fixed knot type (such as the trefoil) grows much slower than the squared writhe of all polygons. We also observe that while the writhe values at a given length and confinement radius are normally distributed, the distribution of the average crossing number values around their mean are not normal, but rather log-normal.
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It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show... more
It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k-thickening or k-stretching, then its Tutte polynomial can be expressed in terms of the Tutte polynomial of the original graph, thus enabling us to compute the Jones polynomials for some (special) large non-alternating knots.
Research Interests: Knots and Signed graphs
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For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula:... more
For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: s...
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For a knot or link K, let L(K) denote the ropelength of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the relationship between L(K) and Cr(K) (or intuitively, the relationship... more
For a knot or link K, let L(K) denote the ropelength of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the relationship between L(K) and Cr(K) (or intuitively, the relationship between the length of a rope needed to tie a particular knot and the complexity of the knot). We show that there exists a constant a &amp;gt; 0 such that if a knot K allows a special knot diagram D (called Conway algebraic knot diagram) with n crossings, then L(K) • a¢n. Furthermore, if D is alternating (but not necessarily reduced and in fact K may not have a minimal alternating diagram that is algebraic), then L(K) • a ¢ Cr(K). The approach used here can be applied to a larger class of knots, namely those formed by replacing single crossings in a Conway algebraic knot diagram by tangles whose crossing number is bounded by a constant. Interestingly, it has been shown by the same authors that the Jones polynomials of these knots can be computed in polynomial time.
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An energy function for smooth knots is defined using the concept of thickness and proved to be a good indicator of the complexity of knots. This energy function has the advantage that it can be measured experimentally. We carry out an... more
An energy function for smooth knots is defined using the concept of thickness and proved to be a good indicator of the complexity of knots. This energy function has the advantage that it can be measured experimentally. We carry out an experiment to measure the energies for knots up to 8 crossings and compare our data with energies obtained elsewhere through numerical simulations.
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Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates... more
Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special
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Research Interests: Random Walks and Knots
... finite. Thus, the expected value of the number of circles in a finite union of cubes is also finite. Consequently, an unsplittable link with an infinite number of circles can only appear if it is in an infinite union of cubes. Hence... more
... finite. Thus, the expected value of the number of circles in a finite union of cubes is also finite. Consequently, an unsplittable link with an infinite number of circles can only appear if it is in an infinite union of cubes. Hence ST&amp;gt; 6C. ...
It has been shown that the smallest knots on the cubic lattice are all trefoils of length 24. In this paper, we show that the number of such unrooted knots on the cubic lattice is 3496.