The Andersen-Kashaev volume conjecture for FAMED geometric triangulations
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arXiv
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| Format: | Preprint |
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2024
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| author | Aribi, Fathi Ben Wong, Ka Ho |
| author_facet | Aribi, Fathi Ben Wong, Ka Ho |
| contents | We investigate the Andersen-Kashaev volume conjecture by introducing the notion of FAMED triangulations, a class of ideal triangulations of $3$-manifolds satisfying certain specific combinatorial properties. For any FAMED triangulation of a one-cusped hyperbolic $3$-manifold $M$ with trivial second homology, we prove the existence of the Jones function in the Teichmüller TQFT of $M$.
For FAMED geometric triangulations of $M$, we establish an asymptotic expansion of the Jones function in terms of the Neumann-Zagier potential function and the 1-loop invariant of Dimofte-Garoufalidis. As a consequence, we prove the Andersen-Kashaev volume conjecture for $M$ and provide new insights for the AJ conjecture for the Teichmüller TQFT developed by Andersen-Malusa. We further discover a new phenomenon: for FAMED geometric triangulations, the partition function in Teichmüller TQFT decays exponentially with decrease rate the hyperbolic volume of a cone structure determined by the prescribed angle structure. This perspective provides a potential application to the Casson conjecture on angle structures.
Expanding the previous result of Guéritaud, Piguet-Nakazawa and the first author and complementing a parallel result of Guilloux and both authors, we prove all the above generalizations of the Andersen-Kashaev volume conjecture for every hyperbolic twist knot and for the first 42,000 hyperbolic knots in $S^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10776 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Andersen-Kashaev volume conjecture for FAMED geometric triangulations Aribi, Fathi Ben Wong, Ka Ho Geometric Topology 57K31, 57K32 We investigate the Andersen-Kashaev volume conjecture by introducing the notion of FAMED triangulations, a class of ideal triangulations of $3$-manifolds satisfying certain specific combinatorial properties. For any FAMED triangulation of a one-cusped hyperbolic $3$-manifold $M$ with trivial second homology, we prove the existence of the Jones function in the Teichmüller TQFT of $M$. For FAMED geometric triangulations of $M$, we establish an asymptotic expansion of the Jones function in terms of the Neumann-Zagier potential function and the 1-loop invariant of Dimofte-Garoufalidis. As a consequence, we prove the Andersen-Kashaev volume conjecture for $M$ and provide new insights for the AJ conjecture for the Teichmüller TQFT developed by Andersen-Malusa. We further discover a new phenomenon: for FAMED geometric triangulations, the partition function in Teichmüller TQFT decays exponentially with decrease rate the hyperbolic volume of a cone structure determined by the prescribed angle structure. This perspective provides a potential application to the Casson conjecture on angle structures. Expanding the previous result of Guéritaud, Piguet-Nakazawa and the first author and complementing a parallel result of Guilloux and both authors, we prove all the above generalizations of the Andersen-Kashaev volume conjecture for every hyperbolic twist knot and for the first 42,000 hyperbolic knots in $S^3$. |
| title | The Andersen-Kashaev volume conjecture for FAMED geometric triangulations |
| topic | Geometric Topology 57K31, 57K32 |
| url | https://arxiv.org/abs/2410.10776 |