Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911377861378048 |
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| author | McPhail-Snyder, Calvin |
| author_facet | McPhail-Snyder, Calvin |
| contents | Hyperbolic structures on link complements (equivalently, representations of the fundamental group into $\operatorname{SL}_2(\mathbb{C})$) can be described algebraically by using the octahedral decomposition determined by a link diagram. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations can be obtained from Kashaev-Reshetikhin's braiding on the Kac-de Concini quantum group $\mathcal{U}_ξ(\mathfrak{sl}_2)$ at a root of unity $ξ$. This braiding gives coordinates on the $\operatorname{SL}_2(\mathbb{C})$ representation variety of a link and our work shows how to interpret these geometrically. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_06042 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$ McPhail-Snyder, Calvin Geometric Topology Quantum Algebra 57K32, 57K10, 20G42 Hyperbolic structures on link complements (equivalently, representations of the fundamental group into $\operatorname{SL}_2(\mathbb{C})$) can be described algebraically by using the octahedral decomposition determined by a link diagram. The decomposition (like any ideal triangulation) gives a set of gluing equations in shape parameters whose solutions are hyperbolic structures. We show that these equations can be obtained from Kashaev-Reshetikhin's braiding on the Kac-de Concini quantum group $\mathcal{U}_ξ(\mathfrak{sl}_2)$ at a root of unity $ξ$. This braiding gives coordinates on the $\operatorname{SL}_2(\mathbb{C})$ representation variety of a link and our work shows how to interpret these geometrically. |
| title | Hyperbolic structures on link complements, octahedral decompositions, and quantum $\mathfrak{sl}_2$ |
| topic | Geometric Topology Quantum Algebra 57K32, 57K10, 20G42 |
| url | https://arxiv.org/abs/2203.06042 |