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7.9 The Kauffman Polynomial

The Kauffman polynomial $F(K)(a,z)$ (see [Ka2]) of a knot or link $K$ is $a^{-w(K)}L(K)$ where $w(L)$ is the writhe of $K$ (see Section 7.5.1) and where $L(K)$ is the regular isotopy invariant defined by the skein relations

$$L(s_+)=aL(s), \qquad L(s_-)=a^{-1}L(s)$$

(here $s$ is a strand and $s_\pm$ is the same strand with a $\pm$ kink added) and

$$L\left(\text{\LARGE\raisebox{-4pt}{$\backoverslash$}}\right) +L\l... ...g$}}\right) +L\left(\text{\LARGE\raisebox{-4pt}{$\hsmoothing$}}\right) \right)$$

and by the initial condition $L(\bigcirc)=1$.

KnotTheory` knows about the Kauffman polynomial:

In[1]

In[2]:= ?Kauffman
Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.

In[3]:= Kauffman::about
The Kauffman program was written by Scott Morrison.

Thus, for example, here's the Kauffman polynomial of the knot $5_2$:

In[4]:=	Kauffman[Knot[5, 2]][a, z]
Out[4]=	2 4 6 5 7 2 2 4 2 6 2 3 3 5 3 -a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z + 2 a z + 7 3 4 4 6 4 > a z + a z + a z

It is well known that the Jones polynomial is related to the Kauffman polynomial via

$$J(L)(q) = (-1)^cL(K)(-q^{-3/4}, q^{1/4}+q^{-1/4}),$$

where $K$ is some knot or link and where $c$ is the number of components of $K$. Let us verify this fact for the torus knot $T(8,3)$:

In[5]:=

K = TorusKnot[8, 3];

In[6]:=	Simplify[{ (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)], Jones[K][q] }]
Out[6]=	7 9 16 7 9 16 {q + q - q , q + q - q }