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

单词

BLM/Ho Polynomial

释义

BLM/Ho Polynomial

A 1-variable unoriented Knot Polynomial . It satisfies

(1)

and the Skein Relationship

(2)

It also satisfies

(3)

where

is the Knot Sum and

(4)

where

is the Mirror Image of

. The BLM/Ho polynomials of Mutant Knots are also identical. Brandt et al. (1986)give a number of interesting properties. For any Link

with

components,

is divisible by

. If

has

components, then the lowest Power of

, and

(5)

where

is the HOMFLY Polynomial. Also, the degree of

is less than the Crossing Number of

. If

is a 2-Bridge Knot, then

(6)

where

(Kanenobu and Sumi 1993).

The Polynomial was subsequently extended to the 2-variable Kauffman PolynomialF, which satisfies

(7)

Brandt et al. (1986) give a listing of

Polynomials for Knots up to 8 crossings and links up to 6 crossings.

References

Brandt, R. D.; Lickorish, W. B. R.; and Millett, K. C. ``A Polynomial Invariant for Unoriented Knots and Links.'' Invent. Math. 84, 563-573, 1986.

Ho, C. F. ``A New Polynomial for Knots and Links--Preliminary Report.'' Abstracts Amer. Math. Soc. 6, 300, 1985.

Kanenobu, T. and Sumi, T. ``Polynomial Invariants of 2-Bridge Knots through 22-Crossings.'' Math. Comput. 60, 771-778 and S17-S28, 1993.

Stoimenow, A. ``Brandt-Lickorish-Millett-Ho Polynomials.'' http://www.informatik.hu-berlin.de/~stoimeno/ptab/blmh10.html.

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

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