The Kauffman Polynomial

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

[math]\displaystyle{ L(s_+)=aL(s), \qquad L(s_-)=a^{-1}L(s) }[/math]

(here [math]\displaystyle{ s }[/math] is a strand and [math]\displaystyle{ s_\pm }[/math] is the same strand with a [math]\displaystyle{ \pm }[/math] kink added) and

[math]\displaystyle{ L(\backoverslash)+L(\slashoverback) = z\left(L(\smoothing)+L(\hsmoothing)\right) }[/math]

and by the initial condition [math]\displaystyle{ L(U)=1 }[/math] where [math]\displaystyle{ U }[/math] is the unknot .

KnotTheory` knows about the Kauffman polynomial:

(For In[1] see Setup)

In[1]:= ?Kauffman Kauffman[K][a, z] computes the Kauffman polynomial of a knot or link K, in the variables a and z.

In[2]:= Kauffman::about The Kauffman program was written by Scott Morrison.

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

In[3]:=	Kauffman[Knot[5, 2]][a, z]
Out[3]=	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

[math]\displaystyle{ J(L)(q) = (-1)^cL(K)(-q^{-3/4},\,q^{1/4}+q^{-1/4}) }[/math],

where [math]\displaystyle{ K }[/math] is some knot or link and where [math]\displaystyle{ c }[/math] is the number of components of [math]\displaystyle{ K }[/math]. Let us verify this fact for the torus knot T(8,3):

In[4]:=

K = TorusKnot[8, 3];

In[5]:=	Simplify[{ (-1)^(Length[Skeleton[K]]-1)Kauffman[K][-q^(-3/4), q^(1/4)+q^(-1/4)], Jones[K][q] }]
Out[5]=	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.