# The Kauffman Polynomial

The Kauffman polynomial ${\displaystyle F(K)(a,z)}$ (see [Kauffman]) of a knot or link ${\displaystyle K}$ is ${\displaystyle a^{-w(K)}L(K)}$ where ${\displaystyle w(L)}$ is the writhe of ${\displaystyle K}$ (see How is the Jones Polynomial Computed?) and where ${\displaystyle L(K)}$ is the regular isotopy invariant defined by the skein relations

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

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

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

and by the initial condition ${\displaystyle L(U)=1}$ where ${\displaystyle U}$ 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

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

where ${\displaystyle K}$ is some knot or link and where ${\displaystyle c}$ is the number of components of ${\displaystyle K}$. 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.