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

Abstract

We show that for any Legendrian link $L$ in the 1-jet space of $S^1$ the 2-graded ruling polynomial, $R_L^2(z)$, is determined by the Thurston–Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover $R_L^2(z)$ 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.