On the S^1 x S^2 HOMFLY-PT invariant and Legendrian links

TL;DR

This paper provides formulas to compute the HOMFLY-PT invariant for links in S^1 x S^2, demonstrating it is a Laurent polynomial, and extends Legendrian link invariants and Thurston-Bennequin number estimates to this setting.

Contribution

It introduces explicit formulas for the HOMFLY-PT invariant in S^1 x S^2 and connects skein modules with Legendrian link invariants, extending known polynomial bounds.

Findings

Invariant is a Laurent polynomial in a and z

Formulas enable computation for any link in S^1 x S^2

Extension of Thurston-Bennequin bounds to Legendrian links in S^1 x S^2

Abstract

In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ which is a rational function in variables $a$ and $s$ and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in $S^1\times S^2$ and shows that the invariant is in fact a Laurent polynomial in $a$ and $z= s -s^{-1}$. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston-Bennequin number to Legendrian links in $S^1\times S^2$ with its tight contact structure.