`$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\LP}{\left(} \newcommand{\RP}{\right)} \newcommand{\LS}{\left\lbrace} \newcommand{\RS}{\right\rbrace} \newcommand{\LA}{\left\langle} \newcommand{\RA}{\right\rangle} \newcommand{\LB}{\left[} \newcommand{\RB}{\right]} \newcommand{\MM}{\ \middle|\ } \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\msr}[1]{m\left(#1\right)} \newcommand{\inv}[1]{#1^{-1}} \newcommand{\bkt}[1]{\LA \img{#1}\RA} \require{color}$`

Lets build some knot polynomials

Skein Relation

$$\Huge{\begin{matrix} \img{/presentations/kauf_bkt/crossing/crossing_un.svg} &\quad\img{/presentations/kauf_bkt/type2/6a.svg} &\quad\img{/presentations/kauf_bkt/type2/6b.svg}\\ + &\quad 0 &\quad\infty\\ \end{matrix}}$$

$$\Huge{\LA \img{/presentations/kauf_bkt/crossing/crossing_un.svg}\RA=A\LA \img{/presentations/kauf_bkt/type2/6a.svg} \RA+B\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA}$$

Link Diagrams

We want a knot invariant. How can we get one?

Check what happens under Reidemeister moves

Type 2

$$\Huge{\LA\img{/presentations/kauf_bkt/type2/1.svg}\RA=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA}$$

$${\small\LA \img{/presentations/kauf_bkt/crossing/crossing_un.svg}\RA=A\LA \img{/presentations/kauf_bkt/type2/6a.svg} \RA+B\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA}$$

$$\huge \begin{aligned} 1.\quad&{\LA \img{/presentations/kauf_bkt/unknot.svg} \RA=1}\\ 2.\quad&{\LA D \sqcup \img{/presentations/kauf_bkt/unknot.svg} \RA=C\LA D\RA} \end{aligned}$$

$$\large\begin{aligned} A\LP A\bkt{/presentations/kauf_bkt/type2/3a.svg}+B\bkt{/presentations/kauf_bkt/type2/4.svg}\RP &+B\LP A\bkt{/presentations/kauf_bkt/type2/6b.svg}+B\bkt{/presentations/kauf_bkt/type2/6a.svg}\RP\\ &=A\LP A\bkt{/presentations/kauf_bkt/type2/6a.svg}+BC\bkt{/presentations/kauf_bkt/type2/6a.svg}\RP\\ &+B\LP A\bkt{/presentations/kauf_bkt/type2/6b.svg}+B\bkt{/presentations/kauf_bkt/type2/6a.svg}\RP\\ &=A^2\bkt{/presentations/kauf_bkt/type2/6a.svg}+ABC\bkt{/presentations/kauf_bkt/type2/6a.svg}\\ &+BA\bkt{/presentations/kauf_bkt/type2/6b.svg}+B^2\bkt{/presentations/kauf_bkt/type2/6a.svg}\\ &=\LP A^2+ABC+B^2\RP\bkt{/presentations/kauf_bkt/type2/6a.svg}\\ &+BA\bkt{/presentations/kauf_bkt/type2/6b.svg} \end{aligned}$$

$$\Large\LP A^2+ABC+B^2\RP\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\bkt{/presentations/kauf_bkt/type2/6b.svg}=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA$$

$$B=\inv{A}$$ $$\Large\begin{aligned} \LP A^2+ABC+B^2\RP\bkt{/presentations/kauf_bkt/type2/6a.svg}+BA\bkt{/presentations/kauf_bkt/type2/6b.svg}&=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA\\ \LP A^2+C+A^{-2}\RP\bkt{/presentations/kauf_bkt/type2/6a.svg}+\bkt{/presentations/kauf_bkt/type2/6b.svg}&=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA\\ \end{aligned}$$

$$C=-A^{-2}-A^2$$ $$\Large\begin{aligned} \LP A^2+C+A^{-2}\RP\bkt{/presentations/kauf_bkt/type2/6a.svg}+\bkt{/presentations/kauf_bkt/type2/6b.svg}&=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA\\ \bkt{/presentations/kauf_bkt/type2/6b.svg}&=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA\\ \end{aligned}$$

Type 2

$$\Huge{\LA\img{/presentations/kauf_bkt/type2/1.svg}\RA=\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA}$$ $$\large \begin{aligned} 1.\quad&\LA \img{/presentations/kauf_bkt/crossing/crossing_un.svg}\RA=A\LA \img{/presentations/kauf_bkt/type2/6a.svg} \RA+\inv{A}\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA\\ 2.\quad&{\LA \img{/presentations/kauf_bkt/unknot.svg} \RA=1}\\ 3.\quad&{\LA D \sqcup \img{/presentations/kauf_bkt/unknot.svg} \RA=\LP-A^{-2}-A^2\RP\LA D\RA} \end{aligned}$$

Type 3

$$\Huge{\LA\img{/presentations/kauf_bkt/type3/1.svg}\RA=\LA\img{/presentations/kauf_bkt/type3/6.svg}\RA}$$

What about Type 1?

More later.

Kauffman Bracket

The Kauffman Bracket is a function from unoriented link diagrams to Laurent polynomials with integer coefficients in an indeterminate $A$.

The Kauffman Bracket maps a diagram $D$ to $$\large{\LA D \RA\in \Z\LB A^{-1},\ A\RB}$$ and is characterized by our three relations

Exercise

$$ \LARGE\begin{aligned} \bkt{/presentations/kauf_bkt/trefoil/trefoil_b.svg} &=A\bkt{/presentations/kauf_bkt/trefoil/trefoil_ba.svg} +A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_bb.svg}\\ &=A\LP A\bkt{/presentations/kauf_bkt/trefoil/trefoil_baa.svg}+ A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_bab.svg}\RP\\ &+A^{-1}\LP A\bkt{/presentations/kauf_bkt/trefoil/trefoil_bba.svg}+ A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_bbb.svg}\RP\\ \end{aligned} $$

$$ \LARGE\begin{aligned} &=A\LP A\LP-A^{-2}-A^2\RP+ A^{-1}\RP\\ &+A^{-1}\LP A+ A^{-1}\LP-A^{-2}-A^2\RP\RP\\ &=-A^4-A^{-4}\\ \end{aligned} $$

1. $\small\LA \img{/presentations/kauf_bkt/crossing/crossing_un.svg}\RA=A\LA \img{/presentations/kauf_bkt/type2/6a.svg} \RA+A^{-1}\LA\img{/presentations/kauf_bkt/type2/6b.svg}\RA$
2. $\small\LA \img{/presentations/kauf_bkt/unknot.svg} \RA=1$
3. $\small \LA D \sqcup \img{/presentations/kauf_bkt/unknot.svg} \RA=\LP-A^{-2}-A^2\RP\LA D\RA$

$$ \LARGE\begin{aligned} \bkt{/presentations/kauf_bkt/trefoil/trefoil_a.svg} &=A\bkt{/presentations/kauf_bkt/trefoil/trefoil_aa.svg} +A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_ab.svg}\\ &=A\LP A\bkt{/presentations/kauf_bkt/trefoil/trefoil_aaa.svg}+ A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_aab.svg}\RP\\ &+A^{-1}\LP A\bkt{/presentations/kauf_bkt/trefoil/trefoil_aba.svg}+ A^{-1}\bkt{/presentations/kauf_bkt/trefoil/trefoil_abb.svg}\RP\\ \end{aligned} $$

$$ \Large\begin{aligned} &=A\LP A\LP A\LP-A^{-2}-A^2\RP^2\RP\right.\\ &\left.+A^{-1}\LP A\LP-A^{-2}-A^2\RP\RP\RP\\ &+A^{-1}\LP A\LP A\LP-A^{-2}-A^2\RP+A^{-1}\RP\RP\\ &=A^6 \end{aligned} $$

$$\Huge{\LA\img{/presentations/kauf_bkt/type1/1b.svg}\RA=-A^{-3}\LA\img{/presentations/kauf_bkt/type1/2a.svg}\RA}$$

Writhe

Orientation of a crossing

1. Positive $+1$

2. Negative $-1$

Writhe of a knot

The writhe $w\LP D\RP$ of a diagram $D$ of an oriented link is the sum of the signs of the crossings of $ D $.

$$\Huge{w\LP D\RP=\text{#}\LP\img{/presentations/kauf_bkt/crossing/crossing_+.svg}\RP-\text{#}\LP\img{/presentations/kauf_bkt/crossing/crossing_-.svg}\RP}$$

Consider

$$\Huge -A^{-3w\LP D\RP}\LA D\RA$$

$$\Large -A^{-3w\LP D\RP}\LA\img{/presentations/kauf_bkt/type1/1b.svg}\RA=\large -A^{-3\LP-1\RP}\LP-A^{-3}\RP\LA\img{/presentations/kauf_bkt/type1/2a.svg}\RA=\LA\img{/presentations/kauf_bkt/type1/2a.svg}\RA $$

Definition of the Jones Polynomial

The Jones Polynomial $V\LP \mathscr{L}\RP$ of an oriented link $\mathscr{L}$ is the Laurent polynomial in $t^{1/2}$, with integer coefficients, defined by $$ V\LP \mathscr{L}\RP=\LP\LP-A\RP^{-3w(D)}\LA D \RA\RP _{t^{1/2}=A^{-2}} $$ where $D$ is any oriented diagram for $\mathscr{L}$.

Detects mirrors

$$-A^{16}+A^{12}+A^{4}$$

$$-A^{-16}+A^{-12}+A^{-4}$$

$$-t^{-4}+t^{-3}+t^{-1}$$

$$-t^{4}+t^{3}+t$$

Worksheet

Sources