We show that for any Legendrian link in the 1-jet space of the 2-graded ruling polynomial, , is determined by the Thurston--Bennequin number and the HOMFLY-PT polynomial. Specifically, we recover 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.
Cite this article
Dan Rutherford, HOMFLY-PT polynomial and normal rulings of Legendrian solid torus links. Quantum Topol. 2 (2011), no. 2, pp. 183–215