The Kauffman Polynomial
The Kauffman polynomial (see [Kauffman]) of a knot or link is where is the writhe of (see How is the Jones Polynomial Computed?) and where is the regular isotopy invariant defined by the skein relations
(here is a strand and is the same strand with a kink added) and
and by the initial condition where is the unknot .
KnotTheory`
knows about the Kauffman polynomial:
(For In[1] see Setup)


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
where is some knot or link and where is the number of components of . 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) 417471.