On The Homflypt Skein Module of S^1 x S^2

Abstract

Let $k$ be a subring of the field of rational functions in $x, v, s$ which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$. 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) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$; (2) $L$ with a positive twist $=(xv^{-1})L$; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ 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 $S(S^1\times D^2)$ given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for $S(S^1\times D^2)$ 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 $S(S^{1} \times S^2)=k$-torsion module $\oplus k$. Here the free part is generated by the empty link $\phi$. In addition, if the elements $s^{2m}-v^{4}$, for $m$ an integer, are invertible in $k$, then $S(S^{1} \times S^2)$ has no torsion. We also obtain some results for more general $k$.