The Kauffman Polynomial

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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

$L(T_{1})+L(T_{2})=z\left(L(T_{3})+L(T_{4})\right)$

(here $T_{1}$, $T_{2}$, $T_{3}$ and $T_{4}$ are , , and , respectively), and by the initial condition $L(U)=1$ where $U$ is the unknot .

{\tt KnotTheory`} knows about the Kauffman polynomial:

\latexhtml{\small (for {\tt In[1]} see Section~\ref{sec:Setup}.)}{\htmlref{{\tt In[1]}}{sec:Setup}} %<* InOut[1] *>

\index{Kauffman, Louis} \index{Morrison, Scott} <* HelpBox[Kauffman] *>

Thus, for example, here's the Kauffman polynomial of the knot \hlink{../Knots/5.2.html}{$5_2$}:

<*InOut@"Kauffman[Knot[5, 2]][a, z]"*> \vskip 6pt

\index{Jones polynomial} \index{Jones@{\tt Jones}} 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 \hlink{../TorusKnots/8.3.html}{$T(8,3)$}:

<*InOut@"K = TorusKnot[8, 3];"*> <*InOut@"Simplify[{\n

 (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)],\n
 Jones[K][q]\n

}]"*>

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