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.
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Transcription
Contents
Construction
A twist knot is obtained by linking together the two ends of a twisted loop. Any number of halftwists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots:
Properties
All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2bridge knot.^{[1]} Of the twist knots, only the unknot and the stevedore knot are slice knots.^{[2]} A twist knot with halftwists has crossing number . All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figureeight knot.
Invariants
The invariants of a twist knot depend on the number of halftwists. 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
References
 ^ Rolfsen, Dale (2003). Knots and links. Providence, R.I: AMS Chelsea Pub. p. 114. ISBN 0821834363.
 ^ Weisstein, Eric W. "Twist Knot". MathWorld.