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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

 <L>(A,B,d)=sum_(sigma)<L|sigma>d^(||sigma||),
(1)

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

 ||sigma||=N_L-1,
(2)

where N_L is the number of loops in sigma.

Letting

B=A^(-1)
(3)
d=-A^2-A^(-2)
(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 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

 <L>(A)=-A^5-A^(-3)+A^(-7)
(5)

(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

 X(A)=(-A^3)^(-w(L))<L>(A),
(6)

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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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.

Referenced on Wolfram|Alpha

Bracket Polynomial

Cite this as:

Weisstein, Eric W. "Bracket Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BracketPolynomial.html

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