An expression for the Homflypt polynomial and some applications

Details An expression for the Homflypt polynomial and some applications An expression for the Homflypt polynomial and some applications

2010-09-26

arxiv.org/abs/1009.5056v2

Mathematics

Geometric Topology

15 pages

Scientific paper

Associated with each oriented link is the two variable Homflypt polynomial. Formulas for the coefficient polynomials of the three lowest v-degrees are presented that shows they are determined by the writhe of any braid diagram for the link, the Conway polynomial for the link, and the remaining coefficient polynomials. This is used to show the Jones and Homflypt polynomials distinguish the same three-braid links. These Homflypt coefficient polynomials in z satisfy a system of linear equations with coefficients in Z[z]. The Conway polynomial is essentially the unique Laurent polynomial that represents such a linear combination and is also a link invariant; any other is merely the product of the Conway polynomial and an arbitrary second polynomial. Two other independent functions that represent such a linear combination are determined by the writhe and are not link invariants.

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