On the Kauffman bracket skein module of (S^1 × S^2) # (S^1 × S^2)

Determining the structure of the Kauffman bracket skein module of all 3-manifolds over the ring of Laurent polynomials ℤ[A^± 1] 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 × S^2) # (S^1 × S^2) over the ring ℤ[A^± 1]. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we also compute the Kauffman bracket skein module of (S^1 × S^2) # (S^1 × D^2). Furthermore, we show that the skein module of (S^1 × S^2) # (S^1 × S^2) does not split into the sum of free and (A^k-A^-k)-torsion modules, for each k≥ 1.