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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

 X_L(A)=(-A^3)^(-w(L))<L>(A),
(1)

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

 X_(L^*)=XL_L(A^(-1)).
(2)

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

 X(A)=V(A^(-4))
(3)

and related to the two-variable Kauffman polynomial F by

 X(A)=F(-A^(-3),A+A^(-1)).
(4)

The Kaufman X-polynomial of the trefoil knot is therefore

 X(A)=A^(-4)+A^(-12)-A^(-16)
(5)

(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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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.

Referenced on Wolfram|Alpha

Kauffman Polynomial X

Cite this as:

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

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