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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Blue_8_1_Knot.png/220px-Blue_8_1_Knot.png)
Construction
editA 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
editAll 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.
Invariants
editThe 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
References
edit- ^ Rolfsen, Dale (2003). Knots and links. Providence, R.I: AMS Chelsea Pub. pp. 114. ISBN 0-8218-3436-3.
- ^ Weisstein, Eric W. "Twist Knot". MathWorld.