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

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We show that for any Legendrian link L in the 1-jet space of S¹ the 2-graded ruling polynomial, R²ₗ(z), is determined by the Thurston-Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover R²ₗ(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.