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

Rhea Palak Bakshi; Seongjeong Kim; Shangjun Shi; Xiao Wang

paper

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

arXiv:2405.04337

Abstract

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.

31 pages, 20 figures

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

Abstract

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