Twist knot

In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.

A twist knot with six half-twists.

Construction

A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots:

• 
  One half-twist
  (trefoil knot, 3₁)
• 
  Two half-twists
  (figure-eight knot, 4₁)
• 
  Three half-twists
  (5₂ knot)
• 
  Four half-twists
  (stevedore knot, 6₁)
• 
  Five half-twists
  (7₂ knot)
• 
  Six half-twists
  (8₁ knot)

Properties

The four half-twist stevedore knot is created by passing the one end of an unknot with four half-twists through the other.

All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2-bridge knot. Of the twist knots, only the unknot and the stevedore knot are slice knots. A twist knot with $n$ half-twists has crossing number $n+2$. All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot.

Invariants

The invariants of a twist knot depend on the number $n$ of half-twists. The Alexander polynomial of a twist knot is given by the formula

$\Delta (t)={\begin{cases}{\frac {n+1}{2}}t-n+{\frac {n+1}{2}}t^{-1}&{\text{if }}n{\text{ is odd}}\\-{\frac {n}{2}}t+(n+1)-{\frac {n}{2}}t^{-1}&{\text{if }}n{\text{ is even,}}\\\end{cases}}$

and the Conway polynomial is

$\nabla (z)={\begin{cases}{\frac {n+1}{2}}z^{2}+1&{\text{if }}n{\text{ is odd}}\\1-{\frac {n}{2}}z^{2}&{\text{if }}n{\text{ is even.}}\\\end{cases}}$

When $n$ is odd, the Jones polynomial is

$V(q)={\frac {1+q^{-2}+q^{-n}-q^{-n-3}}{q+1}},$

and when $n$ is even, it is

$V(q)={\frac {q^{3}+q-q^{3-n}+q^{-n}}{q+1}}.$