Growth of quantum 6j-symbols and applications to the Volume Conjecture

Belletti, Giulio; Detcherry, Renaud; Kalfagianni, Efstratia; Tian, Ye

doi:10.48550/arxiv.1807.03327

Cited by 4 publications

(11 citation statements)

References 12 publications

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“…This conjecture has been verified for the complements of the Borromean rings [DKY18], of the figure eight knot [DKY18], all the hyperbolic integral Dehn surgeries on the figure eight knot [Oht18], and all complements of fundamental shadow links [BDKY18].…”

Section: Introductionmentioning

confidence: 90%

“…In this section we prove Theorem 4.8. The first main tool used is a sharp upper bound on the asymptotic growth of a 6j-symbol [BDKY18].…”

Section: The Fourier Transformmentioning

confidence: 99%

“…(where the sign is chosen so that the color is always even), we have Remark 5.6. There is an overlap between Theorem 5.5 and Theorem 1.1 of [BDKY18]. Some links of Theorem 5.5 are also Fundamental Shadow Links (FSL); namely, those links corresponding to graphs obtained from the tetrahedron by blow-ups.…”

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confidence: 99%

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A maximum volume conjecture for hyperbolic polyhedra

Belletti¹

2020

Preprint

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We propose a volume conjecture for hyperbolic polyhedra that is similar in spirit to the recent volume conjecture by Chen and Yang on the growth of the Turaev-Viro invariants. Using Barrett's Fourier transform we are able to prove this conjecture in a large family of examples. As a consequence of this result, we prove the Turaev-Viro volume conjecture for a new infinite family of hyperbolic manifolds.

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Section: Introductionmentioning

confidence: 90%

“…In this section we prove Theorem 4.8. The first main tool used is a sharp upper bound on the asymptotic growth of a 6j-symbol [BDKY18].…”

Section: The Fourier Transformmentioning

confidence: 99%

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A maximum volume conjecture for hyperbolic polyhedra

Belletti¹

2020

Preprint

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“…Furthermore, all these mapping classes are obtained as monodromies of fibered links in S 3 . In [5] we prove the Turaev-Viro invariants volume conjecture for an infinite family of cusped hyperbolic 3-manifolds. Considering the doubles of these 3-manifolds we obtain an infinite family of closed 3-manifolds M with lT V (M ) > 0.…”

Section: 1mentioning

confidence: 99%

“…In this paper we will be concerned with surfaces with boundary and mapping classes that appear as monodromies of fibered links in S 3 . In [5], with Belletti and Yang, we construct families of 3-manifolds in which the monodromies of all hyperbolic fibered links satisfy the AMU conjecture. In this paper show the following.…”

Section: Introductionmentioning

confidence: 99%

Quantum representations and monodromies of fibered links

Detcherry

Kalfagianni

2019

Advances in Mathematics

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Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants T Vr of hyperbolic 3-manifolds. We show that if the r-growth of |T Vr(M )| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work [11] we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g > n 2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture.We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links.

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Gromov norm and Turaev-Viro invariants of 3-manifolds

Detcherry,

Kalfagianni

2017

Preprint

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We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants T Vr and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold M , with (possibly empty) toroidal boundary, log |T Vr(M )| is bounded above by a function linear in r and whose slope is a positive universal constant times the Gromov norm of M . The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of 6j-symbols.We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has log |T Vr(M )| B r, for some B > 0. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose SO(3) Turaev-Viro invariants grow exponentially. These constructions are essential for the results of [10] where the authors make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations.We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question of the extent to which relations between the invariants T Vr and hyperbolic volume are preserved under Dehn filling.Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm.

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Growth of quantum 6j-symbols and applications to the Volume Conjecture

Cited by 4 publications

References 12 publications

A maximum volume conjecture for hyperbolic polyhedra

A maximum volume conjecture for hyperbolic polyhedra

Quantum representations and monodromies of fibered links

Gromov norm and Turaev-Viro invariants of 3-manifolds

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