Kauffman Polynomial F

A semi-oriented 2-variable knot polynomial defined by

(1)

where  is an oriented link diagram,  is the writhe of ,  is the unoriented diagram corresponding to , and  is the bracket polynomial. It was developed by Kauffman by extending the BLM/Ho polynomial  to two variables, and satisfies

(2)

The Kauffman polynomial is a generalization of the Jones polynomial  since it satisfies

(3)

but its relationship to the HOMFLY polynomial is not well understood. In general, it has more terms than the HOMFLY polynomial, and is therefore more powerful for discriminating knots. It is a semi-oriented polynomial because changing the orientation only changes  by a power of . In particular, suppose  is obtained from  by reversing the orientation of component , then

(4)

where  is the linking number of  with  (Lickorish and Millett 1988).  is unchanged by mutation.

(5)

(6)

M. B. Thistlethwaite has tabulated the Kauffman 2-variable polynomial for knots up to 13 crossings.

REFERENCES:

Lickorish, W. B. R. and Millett, B. R. "The New Polynomial Invariants of Knots and Links." Math. Mag. 61, 1-23, 1988.

Stoimenow, A. "Kauffman Polynomials." https://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/k10.html.