“Bracket Polynomial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Bracket Polynomial

释义

Bracket Polynomial

A one-variable Knot Polynomial related to the Jones Polynomial. The bracket polynomial, however, is not atopological invariant, since it is changed by type I Reidemeister Moves. However, the Spanof the bracket polynomial is a knot invariant. The bracket polynomial is occasionally given the grandiose name RegularIsotopy Invariant. It is defined by

(1)

where

and

are the ``splitting variables,''

runs through all ``states'' of

obtained by Splitting the Link,

is the product of ``splitting labels''corresponding to

, and

(2)

where

is the number of loops in

. Letting

			(3)
			(4)

gives a Knot Polynomial which is invariant under Regular Isotopy, and normalizing gives the Kauffman Polynomial X which is invariant under Ambient Isotopy. The bracket Polynomialof the Unknot is 1. The bracket Polynomial of the Mirror Image

is the same as for

but with

replaced by

. In terms of the one-variable Kauffman Polynomial X, thetwo-variable Kauffman Polynomial F and the Jones Polynomial

			(5)
			(6)
			(7)

where

is the Writhe of

See also Square Bracket Polynomial

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 148-155, 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. 26-29, 1991.

Weisstein, E. W. ``Knots and Links.'' Mathematica notebook Knots.m.

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