Bracket Polynomial

The bracket polynomial is one-variable knot polynomial related to the Jones polynomial. The bracket polynomial, however, is not a topological invariant, since it is changed by type I Reidemeister moves. However, the polynomial span of the bracket polynomial is a knot invariant, as is a normalized form involving the writhe. The bracket polynomial is occasionally given the grandiose name regular isotopy invariant. It is defined by

(1)

where  and  are the "splitting variables,"  runs through all "states" of  obtained by splitting the link,  is the product of "splitting labels" corresponding to , and

(2)

where  is the number of loops in .

Letting

			(3)
			(4)

gives a knot polynomial which is invariant under regular isotopy, and normalizing gives the Kauffman polynomial X which is invariant under ambient isotopy as well. The bracket polynomial of the unknot is 1. The bracket polynomial of the mirror image  is the same as for  but with  replaced by .

For example, the bracket polynomial of the trefoil knot is given by

(5)

(Kauffman 1991, p. 35; Livingston 1993, p. 218; Adams 1994, p. 158 gives a form with  replaced by ).

The so-called normalized bracket polynomial, also called the Kauffman polynomial X, is defined in terms of the bracket polynomial by

(6)

where  is the writhe of . This normalized version is implemented in the Wolfram Language as KnotData[knot, "BracketPolynomial"].

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 148-155 and 157-158, 1994.

Kauffman, L. "New Invariants in the Theory of Knots." Amer. Math. Monthly 95, 195-242, 1988.

Kauffman, L. Knots and Physics. Teaneck, NJ: World Scientific, pp. 25-29, 1991.

Livingston, C. "Kauffman's Bracket Polynomial." Knot Theory. Washington, DC: Math. Assoc. Amer., pp. 217-220, 1993.