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.