  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.
