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

Determining the structure of the Kauffman bracket skein module of all 3-manifolds over the ring of Laurent polynomials ZA^ 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 ZA^ 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.