en>Rgdboer: lk #Functions of 2 × 2 real matrices

2013-12-13T22:41:20Z

lk #Functions of 2 × 2 real matrices

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[[Image:Blue 8_1 Knot.png|thumb|right|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 (topology)|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 knot]]s.

==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:
<gallery>
Image:One-Twist Trefoil.png|One half-twist ([[trefoil knot]])
Image:Blue Figure-Eight Knot.png|Two half-twists ([[Figure-eight knot (mathematics)|figure-eight knot]])
Image:Blue Three-Twist Knot.png|Three half-twists ([[Three-twist knot|52 knot]])
Image:Blue Stevedore Knot.png|Four half-twists ([[Stevedore knot (mathematics)|stevedore knot]])
Image:Blue 7_2 Knot.png|Five half-twists (72 knot)
Image:Blue 8_1 Knot.png|Six half-twists (81 knot)
</gallery>

==Properties==
[[Image:Twist knot Stevedore steps horizontal.png|thumb|300px|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]].<ref>{{cite book |author=Rolfsen, Dale |title=Knots and links |publisher=AMS Chelsea Pub |location=Providence, R.I |year=2003 |pages= 114|isbn=0-8218-3436-3 |oclc= |doi= |accessdate=}}</ref> Of the twist knots, only the [[unknot]] and the [[stevedore knot (mathematics)|stevedore knot]] are [[slice knot]]s.<ref>{{MathWorld|title=Twist Knot|urlname=TwistKnot}}</ref> A twist knot with <math>n</math> half-twists has [[crossing number (knot theory)|crossing number]] <math>n+2</math>. All twist knots are [[invertible knot|invertible]], but the only [[amphichiral knot|amphichiral]] twist knots are the unknot and the [[figure-eight knot (mathematics)|figure-eight knot]].

==Invariants==
The invariants of a twist knot depend on the number <math>n</math> of half-twists. The [[Alexander polynomial]] of a twist knot is given by the formula

:<math>\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}</math>

and the [[Alexander polynomial#Alexander.E2.80.93Conway polynomial|Conway polynomial]] is

:<math>\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}</math>

When <math>n</math> is odd, the [[Jones polynomial]] is

:<math>V(q) = \frac{1 + q^{-2} + q^{-n} - q^{-n-3}}{q+1},</math>

and when <math>n</math> is even, it is

:<math>V(q) = \frac{q^3 + q - q^{3-n} + q^{-n}}{q+1}.</math>

==References==
{{reflist}}

{{Knot theory|state=collapsed}}

[[Category:Twist knots| ]]

Hypercomplex analysis - Revision history

en>Rgdboer: lk #Functions of 2 × 2 real matrices