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 (S¹ × S²) \ # \ (S¹ × S²) 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 (S¹ × S²) \ # \ (S¹ × D²). Furthermore, we show that the skein module of (S¹ × S²) \ # \ (S¹ × S²) does not split into the sum of free and (Aᵏ-A⁻ᵏ)-torsion modules, for each k≥ 1.