Kauffman polynomial

In knot theory, the Kauffman polynomial is a 2-variable knot polynomial due to Louis Kauffman. It is initially defined on a link diagram as

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

Here s is a strand and s r (resp. s ℓ ) 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).