On the Homflypt Skein Module of S1 S2

Let k be a subring of the eld of rational functions in x; v; s which contains x 1 ; v 1 ; s 1. If M is an oriented 3-manifold, let S(M) denote the Hommypt skein module of M over k. This is the free k-module generated by isotopy classes of framed oriented links in M quotiented by the Hommypt skein relations: (1) x ?1 L + ? xL ? = (s ? s ?1)L 0 ; (2) L with a positive twist = (xv ?1)L; (3) L t O = (v?v ?1 s?s ?1)L where O is the unknot. We give two bases for the relative Hommypt skein module of the solid torus with 2 points in the boundary. The rst basis is related to the basis of S(S 1 D 2) given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for S(S 1 D 2) based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements s 2n ? 1, for n a nonzero integer, and the elements s 2m ?v 2 , for any integer m, are invertible in k, then S(S 1 S 2) = k-torsion module k. Here the free part is generated by the empty link. In addition, if the elements s 2m ? v 4 , for m an integer, are invertible in k, then S(S 1 S 2) has no torsion. We also obtain some results for more general k.