Growth of quantum $6j$-symbols and applications to the volume conjecture

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

doi:10.4310/jdg/1645207506

Cited by 17 publications

(47 citation statements)

References 27 publications

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“…More recently, Wong and Yang extended Ohtsuki's result to closed hyperbolic 3-manifolds obtained from rational surgeries along the figure-eight knot in [35]. Belletti, Detcherry, Kalfagianni, and Yang verified the conjecture for the complements of fundamental shadow links in [5]. This is an infinite class of hyperbolic links in connected sums of copies of S 2 × S 1 which was first considered by Constantino and Thurston [8].…”

Section: Introductionmentioning

confidence: 86%

Fundamental shadow links realized as links in $S^3$

Kumar

2020

Preprint

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We use purely topological tools to construct several infinite families of hyperbolic links in the 3-sphere that satisfy the Turaev-Viro invariant volume conjecture posed by Chen and Yang. To show that our links satisfy the volume conjecture, we prove that each has complement homeomorphic to the complement of a fundamental shadow link. These are links in connected sums of copies of S 2 × S 1 for which the conjecture is known due to Belletti, Detcherry, Kalfagianni, and Yang. Our methods also verify the conjecture for several hyperbolic links with crossing number less than twelve. In addition, we show that every link in the 3-sphere is a sublink of a link that satisfies the conjecture.As an application of our results, we extend the class of known examples that satisfy the AMU conjecture on quantum representations of surface mapping class groups. For example, we give explicit elements in the mapping class group of a genus g surface with four boundary components for any g. For this, we use techniques developed by Detcherry and Kalfagianni which relate the Turaev-Viro invariant volume conjecture to the AMU conjecture.

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Section: Introductionmentioning

confidence: 86%

Fundamental shadow links realized as links in $S^3$

Kumar

2020

Preprint

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“…Part (4) follows by [12,Corollary 8.4] and by the main result of [11]. Part (5) follows from [12,Corollary 5.3] and part ( 6) is just a reformulation of part (5). Finally, part (7) follows by [14, Theorem 3.1 and Remark 3.4].…”

Section: Preliminariesmentioning

confidence: 93%

“…An important open problem in quantum topology is the volume conjecture of [8] asserting that for any finite volume hyperbolic 3-manifold M we have lT V (M ) = Vol(M ), which in particular implies that M is q-hyperbolic. A perhaps more robust conjecture, supported by the computations of [8] and the results of [5,11,12,14], is the following. Conjecture 1.5.…”

Section: Introductionmentioning

confidence: 90%

“…Proof. The Turaev-Viro invariants volume conjecture is known for the classes of manifolds in (1)-( 3) above: For figure-eight knot and the Borromean rings complements the conjecture was proved by Detcherry-Kalfagianni-Yang [14] for the integral surgeries on the figure-eight knot by Ohtsuki [32] and for fundamental shadow link complements by Belletti-Detcherry-Kalfagianni-Yang [5]. Part (4) follows by [12,Corollary 8.4] and by the main result of [11].…”

Section: Preliminariesmentioning

confidence: 99%

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Cosets of monodromies and quantum representations

Detcherry¹,

Kalfagianni²

2020

Preprint

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We use geometric methods to show that given any 3-manifold M , and g a sufficiently large integer, the mapping class group Mod(Σ g,1 ) contains a coset of an abelian subgroup of rank g 2 , consisting of pseudo-Anosov monodromies of open-book decompositions in M. We prove a similar result for rank two free cosets of Mod(Σ g,1 ). These results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.

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“…This result is proved in [4, Lemma 3.5]. To be precise, the authors of [4] studied the maximum of V on boundary points of B H , the non-smooth points and the critical points of the interior smooth points. From this, they proved that V attains its maximum at the unique maximum point (α 1 , .…”

Section: Convexity and Preliminary Estimatementioning

confidence: 94%

On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs

Pandey¹,

Wong²

2021

Preprint

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We verify the leading order term in the asymptotic expansion conjecture of the relative Reshetikhin-Turaev invariants proposed in [56] for all pairs (M, L) satisfying the properties that M L is homeomorphic to some fundamental shadow link complement and the 3-manifold M is obtained by doing rational Dehn filling on some boundary components of the fundamental shadow link complement, under the assumptions that the denominator of the surgery coefficients are odd and the cone angles are sufficiently small. In particular, the asymptotics of the invariants captures the complex volume and the twisted Reidemeister torsion of the manifold M L associated with the hyperbolic cone structure determined by the sequence of colorings of the framed link L.

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Growth of quantum $6j$-symbols and applications to the volume conjecture

Cited by 17 publications

References 27 publications

Fundamental shadow links realized as links in $S^3$

Fundamental shadow links realized as links in $S^3$

Cosets of monodromies and quantum representations

On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs

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