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


where A and B are the "splitting variables," sigma runs through all "states" of L obtained by splitting the link, <L|sigma> is the product of "splitting labels" corresponding to sigma, and


where N_L is the number of loops in sigma.



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 K^* is the same as for K but with A replaced by A^(-1).

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


(Kauffman 1991, p. 35; Livingston 1993, p. 218; Adams 1994, p. 158 gives a form with A replaced by A^(-1)).

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


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

See also

Jones Polynomial, Kauffman Polynomial X, Square Bracket Polynomial

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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.

Referenced on Wolfram|Alpha

Bracket Polynomial

Cite this as:

Weisstein, Eric W. "Bracket Polynomial." From MathWorld--A Wolfram Web Resource.

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