Kauffman Polynomial F

A semi-oriented 2-variable Knot Polynomial defined by

$$F_L(a,z)=a^{-w(L)} \left\langle{\vert L\vert}\right\rangle{},$$ (1)

where $L$ is an oriented Link Diagram, $w(L)$ is the Writhe of $L$, $\vert L\vert$ is the unoriented diagram corresponding to $L$, and $\left\langle{L}\right\rangle{}$ is the Bracket Polynomial. It was developed by Kauffman by extending the BLM/Ho Polynomial $Q$ to two variables, and satisfies

$$F(1,x)=Q(x).$$ (2)

The Kauffman Polynomial is a generalization of the Jones Polynomial $V(t)$ since it satisfies

$$V(t)=F(-t^{-3/4},t^{-1/4}+t^{1/4}),$$ (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 $F$ by a Power of $a$. In particular, suppose $L^*$ is obtained from $L$ by reversing the orientation of component $k$, then

$$F_{L^*}=a^{4\lambda}F_L,$$ (4)

where $\lambda$ is the Linking Number of $k$ with $L-k$ (Lickorish and Millett 1988). $F$ is unchanged by Mutation.

$$F_{{L_1}+F_{L_2}}=F(L_1)F(L_2)$$ (5)

$$F_{L_1\cup L_2}=[(a^{-1}+a)x^{-1}-1]F_{L_1}F_{L_2}.$$ (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.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/k10.html.

Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.