Kauffman Polynomial X

The Kauffman X-polynomial, also called the normalized bracket polynomial, is a 1-variable knot polynomial denoted X (Adams 1994, p. 153), L (Kauffman 1991, p. 33), or F (Livingston 1993, p. 219), and defined for a link L by


where <L> is the bracket polynomial and w(L) is the writhe of L (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


It is identical to the Jones polynomial V(t) with the change of variable


and related to the two-variable Kauffman polynomial F by


The Kaufman X-polynomial of the trefoil knot is therefore


(Kaufmann 1991, p. 35). The following table summarizes the polynomials for named knots.

See also

Bracket Polynomial, Kauffman Polynomial F, Jones Polynomial, Knot Invariant, Knot 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, 1994.Kauffman, L. H. Knots and Physics. Singapore: World Scientific, p. 33, 1991.Livingston, C. Knot Theory. Washington, DC: Math. Assoc. Amer., 1993.

Referenced on Wolfram|Alpha

Kauffman Polynomial X

Cite this as:

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

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