On the HOMFLYPT skein module of S1×S2 via braids

In this paper we study the HOMFLYPT skein module of S1×S2≅L(0,1), denoted S(S1×S2), using braid-theoretic techniques. We extend the Lambropoulou invariant, X, for links in the solid torus ST to links in S1×S2, by solving an infinite system of equations of the form Xa^=Xbbm(a)^, where bbm(a) denotes all possible band moves applied to a, for all a in a basis of S(ST). We show that the free part of S(S1×S2) is generated by the empty link, while all other elements lies in the torsion submodule.