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(T_1)+L(T_2) = z\left(L(T_3)+L(T_4)\right) }[/math]

(here [math]\displaystyle{ T_1 }[/math], [math]\displaystyle{ T_2 }[/math], [math]\displaystyle{ T_3 }[/math] and [math]\displaystyle{ T_4 }[/math] are , , and , respectively), 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)

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

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):

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