A semi-oriented 2-variable Knot Polynomial defined by
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
The Kauffman Polynomial is a generalization of the Jones Polynomial since it satisfies
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
where is the Linking Number of with (Lickorish and Millett 1988). is unchanged by
M. B. Thistlethwaite has tabulated the Kauffman 2-variable Polynomial for Knots up to 13 crossings.
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.
© 1996-9 Eric W. Weisstein