The Kauffman 
-polynomial,
also called the normalized bracket polynomial, is a 1-variable knot
polynomial denoted 
(Adams 1994, p. 153), 
(Kauffman 1991, p. 33), or 
(Livingston 1993, p. 219), and defined for a link 
by
 |
(1)
|
where 
is the bracket polynomial and 
is the writhe of 
(Kauffman 1991, p. 33; Adams 1994, p. 153). It is
implemented in the Wolfram Language
as KnotData[knot,
"BracketPolynomial"].
This polynomial is invariant under ambient isotopy, and relates mirror images by
 |
(2)
|
It is identical to the Jones polynomial 
with the change of variable
 |
(3)
|
and related to the two-variable Kauffman polynomial F by
 |
(4)
|
The Kaufman 
-polynomial
of the trefoil knot is therefore
 |
(5)
|
(Kaufmann 1991, p. 35). The following table summarizes the polynomials for named knots.
-polynomial
