Kauffman polynomial

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 theoriesfor 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.)

References

*Louis Kauffman, "On Knots", (1987), ISBN 0-691-08435-1

External links

* [http://eom.springer.de/k/k120040.htm Springer EoM entry for Kauffman polynomial]
* [http://katlas.math.toronto.edu/wiki/The_Kauffman_Polynomial Knot Atlas entry for Kauffman polynomial]