On The Homflypt Skein Module of S¹ x S²

Original · EN

Let k be a subring of the field 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 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) x⁻¹L₊-xL₋=(s-s⁻¹)L₀; (2) L with a positive twist =(xv⁻¹)L; (3) L O=(v-v⁻¹s-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(S¹× D²) given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for S(S¹× D²) based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements s²ⁿ-1, for n a nonzero integer, and the elements s²ᵐ-v², for any integer m, are invertible in k, then S(S¹ × S²)=k-torsion module ⊕ k. Here the free part is generated by the empty link ϕ. In addition, if the elements s²ᵐ-v⁴, for m an integer, are invertible in k, then S(S¹ × S²) has no torsion. We also obtain some results for more general k.