Twist knot

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A twist knot with six half-twists.

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 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:


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.[1] Of the twist knots, only the unknot and the stevedore knot are slice knots.[2] A twist knot with half-twists has crossing number . All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot.


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

and the Conway polynomial is

When is odd, the Jones polynomial is

and when is even, it is


  1. ^ Rolfsen, Dale (2003). Knots and links. Providence, R.I: AMS Chelsea Pub. p. 114. ISBN 0-8218-3436-3. 
  2. ^ Weisstein, Eric W. "Twist Knot". MathWorld.