Kauffman polynomial

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The Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as

$F(K)(a,z)=a^{-w(K)}L(K)\,$

where $w (K)$ is the writhe of the link diagram and $L (K)$ is a polynomial in a and z defined on diagrams by the following properties:

$L (O) = 1$ (O is the unknot)
$L(s_r)=aL(s), \qquad L(s_\ell)=a^{-1}L(s).$
L is unchanged under type II and III Reidemeister moves

Here $s$ is a strand and $s r$ (resp. $s_\ell$ ) is the same strand with a right-handed (resp. left-handed) curl added (using a type I Reidemeister move).

Additionally L must satisfy Kauffman's skein relation:

The pictures represent the L polynomial of the diagrams which differ inside a disc as shown but are identical outside.

Kauffman showed that L exists and is a regular isotopy invariant of unoriented links. It follows easily that F is an ambient isotopy invariant of oriented links.

The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial. The Kauffman polynomial is related to Chern-Simons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to Chern-Simons gauge theories for SU(N) (see Witten's article "Quantum field theory and the Jones polynomial", in Commun. Math. Phys.)