The Kauffman Bracket Skein Module at an irreducible representation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929257767239680 |
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| author | Farajzadeh-Tehrani, Mohammad Frohman, Charles Kania-Bartoszynska, Joanna |
| author_facet | Farajzadeh-Tehrani, Mohammad Frohman, Charles Kania-Bartoszynska, Joanna |
| contents | In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by $4$, is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Kauffman Bracket Skein Module at an irreducible representation Farajzadeh-Tehrani, Mohammad Frohman, Charles Kania-Bartoszynska, Joanna Geometric Topology Quantum Algebra 57K31 In this paper, we study the Kauffman bracket skein module of closed oriented three-manifolds at a non-multiple-of-four roots of unity. Our main result establishes that the localization of this module at a maximal ideal, which corresponds to an irreducible representation of the fundamental group of the manifold, forms a one-dimensional free module over the localized unreduced coordinate ring of the character variety. We apply this by proving that the dimension of the skein module of a homology sphere with finite character variety, when the order of the root of unity is not divisible by $4$, is greater than or equal to the dimension of the unreduced coordinate ring of the character variety. This leads to a computation of the dimension of the skein module with coefficients in rational functions for homology spheres with tame universal skein module. |
| title | The Kauffman Bracket Skein Module at an irreducible representation |
| topic | Geometric Topology Quantum Algebra 57K31 |
| url | https://arxiv.org/abs/2402.17037 |