A quantization of the $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant of tangle exteriors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916014534426624 |
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| author | McPhail-Snyder, Calvin |
| author_facet | McPhail-Snyder, Calvin |
| contents | We define a sequence of invariants $\mathcal{Z}_{N}^ψ$ of tangles with flat $\mathfrak{sl}_{2}$ connections (i.e. hyperbolic structures) on their complements. These can be interpreted as a geometric twist of the Kashaev invariant or as a quantization of the $\operatorname{SL}_{2}(\mathbb{C})$ Chern-Simons invariant. To support the second interpretation we give a new description $\mathcal{I}^ψ$ of the Chern-Simons invariant of a tangle exterior. $\mathcal{Z}_{N}^ψ$ directly recovers $\mathcal{I}^ψ$ when $N = 1$. We build $\mathcal{Z}_{N}^ψ$ using modules over unrestricted quantum $\mathfrak{sl}_{2}$ at a root of unity and the holonomy $R$-matrices previously constructed by the author and Reshetikhin (arXiv:2509.02354). Unlike most previous constructions of geometric quantum invariants $\mathcal{Z}_{N}^ψ$ is defined without any phase ambiguity. It is natural to conjecture that $\mathcal{Z}_{N}^ψ$ is related to the quantization of Chern-Simons theory with complex, noncompact gauge group $\operatorname{SL}_{2}(\mathbb{C})$ and we discuss how to interpret our results in this context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantization of the $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant of tangle exteriors McPhail-Snyder, Calvin Quantum Algebra Geometric Topology 57K16 (Primary), 57K32, 58J28 (Secondary) We define a sequence of invariants $\mathcal{Z}_{N}^ψ$ of tangles with flat $\mathfrak{sl}_{2}$ connections (i.e. hyperbolic structures) on their complements. These can be interpreted as a geometric twist of the Kashaev invariant or as a quantization of the $\operatorname{SL}_{2}(\mathbb{C})$ Chern-Simons invariant. To support the second interpretation we give a new description $\mathcal{I}^ψ$ of the Chern-Simons invariant of a tangle exterior. $\mathcal{Z}_{N}^ψ$ directly recovers $\mathcal{I}^ψ$ when $N = 1$. We build $\mathcal{Z}_{N}^ψ$ using modules over unrestricted quantum $\mathfrak{sl}_{2}$ at a root of unity and the holonomy $R$-matrices previously constructed by the author and Reshetikhin (arXiv:2509.02354). Unlike most previous constructions of geometric quantum invariants $\mathcal{Z}_{N}^ψ$ is defined without any phase ambiguity. It is natural to conjecture that $\mathcal{Z}_{N}^ψ$ is related to the quantization of Chern-Simons theory with complex, noncompact gauge group $\operatorname{SL}_{2}(\mathbb{C})$ and we discuss how to interpret our results in this context. |
| title | A quantization of the $\operatorname{SL}_2(\mathbb{C})$ Chern-Simons invariant of tangle exteriors |
| topic | Quantum Algebra Geometric Topology 57K16 (Primary), 57K32, 58J28 (Secondary) |
| url | https://arxiv.org/abs/2509.02365 |