An expression for the Homflypt polynomial and some applications

TL;DR

This paper derives new expressions for the Homflypt polynomial's coefficients using the Morton-Franks-Williams inequality, revealing relations among invariants and showing that Jones and Homflypt polynomials distinguish the same three-braid links.

Contribution

It provides explicit formulas for the first three Laurent coefficients of the Homflypt polynomial in terms of other invariants, enhancing understanding of link invariants.

Findings

Derived expressions for the first three Laurent coefficients of the Homflypt polynomial.

Established relations between the Homflypt polynomial coefficients and link invariants.

Showed that Jones and Homflypt polynomials distinguish the same three-braid links.

Abstract

Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a number of interesting relations. One consequence is an expression for the first three Laurent coefficient polynomials in z as a function of the other coefficient polynomials and three link invariants: the minimum v-degree and v-span of the Homflypt polynomial, and the Conway polynomial. These expressions are used to derive additional properties of the Homflypt polynomial for general n-braid links. One specific result is that the Jones and Homflypt polynomials distinguish the same three-braid links.