# The Kauffman Polynomial

The Kauffman polynomial $F(K)(a,z)$ (see [Kauffman]) of a knot or link $K$ is $a^{-w(K)}L(K)$ where $w(L)$ is the writhe of $K$ (see How is the Jones Polynomial Computed?) 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

$L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)$

and by the initial condition $L(U)=1$ where $U$ is the unknot .

KnotTheory knows about the Kauffman polynomial:

(For In[1] see Setup)

 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 polynomial 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 -a + a + a - 2 a z - 2 a z + a z - a z - 2 a z + a z + 5 3 7 3 4 4 6 4 2 a z + a z + a z + a z

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

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

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 }`

[Kauffman] ^  L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 312 (1990) 417-471.