On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917952575504384 |
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| author | Bakshi, Rhea Palak Kim, Seongjeong Shi, Shangjun Wang, Xiao |
| author_facet | Bakshi, Rhea Palak Kim, Seongjeong Shi, Shangjun Wang, Xiao |
| contents | Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ 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 $\mathbb Z[A^{\pm 1}]$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_04337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ Bakshi, Rhea Palak Kim, Seongjeong Shi, Shangjun Wang, Xiao Geometric Topology Quantum Algebra 57K31, 57K10 Determining the structure of the Kauffman bracket skein module of all $3$-manifolds over the ring of Laurent polynomials $\mathbb Z[A^{\pm 1}]$ 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 $\mathbb Z[A^{\pm 1}]$. 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. |
| title | On the Kauffman bracket skein module of $(S^1 \times S^2) \ \# \ (S^1 \times S^2)$ |
| topic | Geometric Topology Quantum Algebra 57K31, 57K10 |
| url | https://arxiv.org/abs/2405.04337 |