Volume Conjecture and quantum hyperbolic invariants: the figure eight knot complement
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911601808900096 |
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| author | Baseilhac, Stephane Aribi, Fathi Ben |
| author_facet | Baseilhac, Stephane Aribi, Fathi Ben |
| contents | We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement $M$. We show that it is rigid, in the sense that it does not depend on the choice of holonomy representation of $M$, and it is either $0$ or equal to the hyperbolic volume of $M$ divided by $2π$, depending on a parity condition satisfied by logarithms of the holonomy eigenvalues on the canonical longitude, where the logarithms are parameters of the QHI of $M$. Along the way we also survey some relevant general features of the QHI. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_16077 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Volume Conjecture and quantum hyperbolic invariants: the figure eight knot complement Baseilhac, Stephane Aribi, Fathi Ben Geometric Topology 57K16, 57K32, 57R05 We compute the real part of the semi-classical limit of the sequence of quantum hyperbolic invariants (QHI) of the figure-eight knot complement $M$. We show that it is rigid, in the sense that it does not depend on the choice of holonomy representation of $M$, and it is either $0$ or equal to the hyperbolic volume of $M$ divided by $2π$, depending on a parity condition satisfied by logarithms of the holonomy eigenvalues on the canonical longitude, where the logarithms are parameters of the QHI of $M$. Along the way we also survey some relevant general features of the QHI. |
| title | Volume Conjecture and quantum hyperbolic invariants: the figure eight knot complement |
| topic | Geometric Topology 57K16, 57K32, 57R05 |
| url | https://arxiv.org/abs/2604.16077 |