On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$

Determining the structure of the Kauffman bracket skein module of all $3$ -manifolds over the ring of Laurent polynomials $Z [A^{\pm 1}]$ 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 $(S^{1} \times S^{2}) # (S^{1} \times S^{2})$ over the ring $Z [A^{\pm 1}]$ . 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.

On the Kauffman bracket skein module of $(S^{1} \times S^{2}) # (S^{1} \times S^{2})$

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On the Kauffman bracket skein module of $(S^{1} \times S^{2}) # (S^{1} \times S^{2})$

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

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

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On the Kauffman bracket skein module of $(S^{1} \times S^{2}) # (S^{1} \times S^{2})$

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On the Kauffman bracket skein module of $(S^{1} \times S^{2}) # (S^{1} \times S^{2})$

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