ON THE S1 × S2 HOMFLY-PT INVARIANT AND LEGENDRIAN LINKS

In [On the HOMFLY-PT skein module of S1 × S2, Math. Z. 237(4) (2001) 769–814], Gilmer and Zhong established the existence of an invariant for links in S1 × S2 which is a rational function in variables a and s and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in S1 × S2 and shows that the invariant is in fact a Laurent polynomial in a and z = s – s-1. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legendrian links. As a corollary, we extend HOMFLY-PT polynomial estimates for the Thurston–Bennequin number to Legendrian links in S1 × S2 with its tight contact structure.