On the Kauffman bracket skein module of (S1 × S2) # (S1 × S2)

Abstract

Determining the structure of the Kauffman bracket skein module of all 3-manifolds over the ring of Laurent polynomials $Z\left[A^{\pm 1}\right]$ is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the 3-manifold $(S^1\times S^2)\#(S^1\times S^2)$ over the ring $Z\left[A^{\pm 1}\right]$. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we compute the Kauffman bracket skein module of $(S^1\times S^2)\#(S^1\times D^2)$. We also show that the skein module of $(S^1\times S^2)\#(S^1\times S^2)$ does not split into the sum of free and torsion submodules. Furthermore, we illustrate two families of torsion elements in this skein module.