TOPICS

# 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.

Kauffman Polynomial X

## Explore with Wolfram|Alpha

More things to try:

## 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." http://www.ms.u-tokyo.ac.jp/~stoimeno/ptab/k10.html.

## Referenced on Wolfram|Alpha

Kauffman Polynomial F

## Cite this as:

Weisstein, Eric W. "Kauffman Polynomial F." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KauffmanPolynomialF.html