The Kauffman Polynomial

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The Kauffman polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(K)(a,z)} (see [Kauffman]) of a knot or link Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-w(K)}L(K)} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(L)} is the writhe of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} (see How is the Jones Polynomial Computed?) and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(K)} is the regular isotopy invariant defined by the skein relations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(s_+)=aL(s), \qquad L(s_-)=a^{-1}L(s)}

(here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s} is a strand and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_\pm} is the same strand with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm} kink added) and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right)}

and by the initial condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L(U)=1} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

5_2

T(8,3)

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J(L)(q) = (-1)^{c+1}L(K)(-q^{-3/4},\,q^{1/4}+q^{-1/4})} ,

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} is some knot or link and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is the number of components of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.