Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911472685154304 |
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| author | Wong, Ka Ho |
| author_facet | Wong, Ka Ho |
| contents | We introduce a generalized FAMED property for ideal triangulations of hyperbolic knot complements in $\mathbb{S}^3$. Given a hyperbolic knot $K$ in $\mathbb{S}^3$ and a semi-geometric triangulation $X$ of $\mathbb{S}^3 \setminus K$ that is generalized FAMED with respect to the longitude. We prove that in the semi-classical limit $\hbar \to 0^+$, for any angle structure $α$, the partition function $\mathscr{Z}_\hbar(X,α)$ in Teichmüller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $α$, and that the 1-loop invariant of Dimofte-Garoufalidis emerges as the 1-loop term. With additional combinatorial conditions on the triangulations, we prove the existence of the Jones function and show that its decay rate is governed by the Neumann-Zagier potential function. In particular, the Andersen-Kashaev volume conjecture holds for every hyperbolic knot whose complement admits such kinds of triangulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23198 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations Wong, Ka Ho Geometric Topology Mathematical Physics 57K31, 57K32 We introduce a generalized FAMED property for ideal triangulations of hyperbolic knot complements in $\mathbb{S}^3$. Given a hyperbolic knot $K$ in $\mathbb{S}^3$ and a semi-geometric triangulation $X$ of $\mathbb{S}^3 \setminus K$ that is generalized FAMED with respect to the longitude. We prove that in the semi-classical limit $\hbar \to 0^+$, for any angle structure $α$, the partition function $\mathscr{Z}_\hbar(X,α)$ in Teichmüller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $α$, and that the 1-loop invariant of Dimofte-Garoufalidis emerges as the 1-loop term. With additional combinatorial conditions on the triangulations, we prove the existence of the Jones function and show that its decay rate is governed by the Neumann-Zagier potential function. In particular, the Andersen-Kashaev volume conjecture holds for every hyperbolic knot whose complement admits such kinds of triangulations. |
| title | Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulations |
| topic | Geometric Topology Mathematical Physics 57K31, 57K32 |
| url | https://arxiv.org/abs/2512.23198 |