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

(1)

where
is the bracket polynomial and is the writhe of (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

(2)

It is identical to the Jones polynomial with the change of variable

(3)

and related to the two-variable Kauffman polynomial
F by

(4)

The Kaufman -polynomial
of the trefoil knot is therefore

(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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