HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links

TL;DR

This paper establishes a relationship between the HOMFLY-PT polynomial, Thurston-Bennequin number, and 2-graded ruling polynomials for Legendrian links in the solid torus, providing new tools for link classification.

Contribution

It demonstrates that the 2-graded ruling polynomial is determined by the HOMFLY-PT polynomial and Thurston-Bennequin number, and interprets a specialization as an inner product in the skein module.

Findings

The 2-graded ruling polynomial is recoverable from the HOMFLY-PT polynomial.

Specialization of the HOMFLY-PT polynomial corresponds to an inner product on symmetric functions.

0-graded ruling polynomials distinguish many Legendrian links with same classical invariants.

Abstract

We show that for any Legendrian link $L$ in the $1$-jet space of $S^1$ the $2$-graded ruling polynomial, $R^2_L(z)$, is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R^2_L(z)$ as a coefficient of a particular specialization of the HOMFLY-PT polynomial. Furthermore, we show that this specialization may be interpreted as the standard inner product on the algebra of symmetric functions that is often identified with a certain subalgebra of the HOMFLY-PT skein module of the solid torus. In contrast to the $2$-graded case, we are able to use $0$-graded ruling polynomials to distinguish many homotopically non-trivial Legendrian links with identical classical invariants.