Kauffman Polynomial X
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.
| knot |
Kaufman  -polynomial |
| figure eight knot |
 |
| Miller Institute knot |
 |
| Perko pair |
 |
| Solomon's seal knot |
 |
| stevedore's knot |
 |
| trefoil knot |
 |
| unknot |
1 |
REFERENCES:
Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, 1994.
Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 33, 1991.
Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.
