The Kauffman polynomial 
is 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, if 
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 knot mutation.
For the connected sum and disjoint union, respectively, it satisfies
 |
(5)
|
![F_(L_1 union L_2)=[(a^(-1)+a)x^(-1)-1]F_(L_1)F_(L_2).](/images/equations/KauffmanPolynomialF/NumberedEquation6.svg) |
(6)
|
M. B. Thistlethwaite has tabulated the Kauffman 2-variable polynomial for knots up to 13 crossings.
