On the Homflypt Skein Module Of

Let k be a subring of the field of rational functions in x, v, s which contains x, v, s. If M is an oriented 3-manifold, let S(M) denote the Homflypt 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 Homflypt skein relations: (1) xL+ − xL− = (s − s)L0; (2) L with a positive twist = (xv)L; (3) L ⊔ O = ( v−v −1 s−s−1 )L where O is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of S(S × D) given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for S(S ×D) based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements s − 1, for n a nonzero integer, and the elements s−v, for any integer m, are invertible in k, then S(S ×S) = k-torsion module ⊕k. Here the free part is generated by the empty link φ. In addition, if the elements s − v, for m an integer, are invertible in k, then S(S × S) has no torsion. We also obtain some results for more general k.