ScalingStacks

In this section we review the Kauffman bracket and its relation to the Jones polynomial following Kauffman [Ka]. Fix an orientation of the 3-space $\mathbb{R}^{3}$. A plane projection $D$ of an oriented link $L$ in $\mathbb{R}^{3}$ is called generic if it has no triple intersections, no tangencies and no cusps. In this paper by a plane projection we will mean a generic plane projection. Given a plane projection $D,$ we assign a Laurent polynomial $<D>\in\mathbb{Z}[q,q^{-1}]$ to $D$ by the following rules

1. 1.

   A simple closed loop evaluates to $q+q^{-1}:$

   
2. 2.

   Each over and undercrossing is a linear combination of two simple resolutions of this crossing:

   
3. 3.

   $<D_{1}\bigsqcup D_{2}>=<D_{1}><D_{2}>$ where $<D_{1}\bigsqcup D_{2}>$ stands for the disjoint union of the diagrams $D_{1}$ and $D_{2}.$

From these rules we deduce that

Curves of the diagram $D$ inherit orientations from that of $L.$ Let $x(D)$ be the number of double points in the diagram $D$ that look like

and $y(D)$ the number of double points that look like

Then the quantity

does not depend on the choice of a diagram $D$ of the oriented link $L$ and is an invariant of $L.$ We denote this invariant by $K(L).$ Up to a simple normalization, $K(L)$ is the Kauffman bracket of link $L$ and equal to the Jones polynomial of $L.$ The Kauffman bracket, $f[L],$ as defined in [Ka], is a Laurent polynomial in an indeterminate $A$ (this $A$ has no relation to the algebra $A$ in Section 2.2 of this paper). One easily sees that setting our $q$ to $-A^{-2}$ and dividing by $(-A^{2}-A^{-2})$ we get $f[L]$:

In this paper we will call $K(L)$ the scaled Kauffman bracket.

Let $L_{1},L_{2}$ and $L_{3}$ be three oriented links that differ as shown below.

The rules for computing the Kauffman bracket imply

Moreover, $K(L)=q+q^{-1}$ if $L$ is the unknot.

The Jones polynomial $V(L)$ of an oriented link $L$ is determined by two properties:

1. 1.

   The Jones polynomial of the unknot is $1.$

2. 2.

   For oriented links $L_{1},L_{2},L_{3}$ as above

   $$t^{-1}V(L_{1})-tV(L_{2})=(\sqrt{t}-\frac{1}{\sqrt{t}})V(L_{3})$$

   (24)

Therefore, the scaled Kauffman bracket and the Jones polynomial are related by