On torsion in the Kauffman bracket skein module of $3$-manifolds
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914848673103872 |
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| author | Belletti, Giulio Detcherry, Renaud |
| author_facet | Belletti, Giulio Detcherry, Renaud |
| contents | We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\mathcal{S}(M)$ has torsion as a $\mathbb{Z}[A^{\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\mathcal{S}(\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_17454 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On torsion in the Kauffman bracket skein module of $3$-manifolds Belletti, Giulio Detcherry, Renaud Geometric Topology Quantum Algebra 57K31 We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\mathcal{S}(M)$ has torsion as a $\mathbb{Z}[A^{\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\mathcal{S}(\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface. |
| title | On torsion in the Kauffman bracket skein module of $3$-manifolds |
| topic | Geometric Topology Quantum Algebra 57K31 |
| url | https://arxiv.org/abs/2406.17454 |