A maximum volume conjecture for hyperbolic polyhedra

Belletti, Giulio

doi:10.48550/arxiv.2002.01904

Cited by 5 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall the projective model for hyperbolic space H 3 ⊆ R 3 ⊆ RP 3 where H 3 is the unit ball of R 3 (for the basic definitions see for example [2]). Notice that for convenience we have picked an affine chart R 3 ⊆ RP 3 , so that it always make sense to speak of segments between two points, half spaces, etcetera; this choice is inconsequential, up to isometry. Isometries, in this model, correspond to projective transformations that preserve the unit sphere.…”

Section: Generalized Hyperbolic Polyhedramentioning

confidence: 99%

“…Definition 2.1. A projective polyhedron in RP 3 is a non-degenerate convex polyhedron in some affine chart of RP 3 . Alternatively, it is the closure of a connected component of the complement of finitely many planes in RP 3 that does not contain any projective line.…”

Section: Generalized Hyperbolic Polyhedramentioning

confidence: 99%

“…Let P n be a sequence of proper or almost proper polyhedra with 1-skeleton Γ and with angles bounded away from 0 and π, and suppose P n converges as n → +∞ to the projective polyhedron P * . Further suppose that all the vertices of P n also converge, and a sequence of vertices v n ∈ P n converges to v ∈ ∂H 3 . Let e 1 , .…”

Section: Proof Of Proposition 43mentioning

confidence: 99%

“…To see this, remember that we only need to show that every edge of Q intersects H 3 . Suppose that an edge e of Q is instead tangent to ∂H 3 . By assumption neither of the endpoints of e can lie on ∂H 3 , hence they must lie outside of H 3 .…”

Section: Case 1 Every Limit Edge E ∞mentioning

confidence: 99%

“…In [3], the author introduces a "Turaev-Viro" invariant of graphs Γ ⊆ S 3 , denoted with T V (Γ), and proposes the following conjecture on its asymptotic behavior in the case where Γ is planar and 3-connected (which is equivalent to saying that Γ is the 1-skeleton of some polyhedron).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The maximum volume of hyperbolic polyhedra

Belletti¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

We study the supremum of the volume of hyperbolic polyhedra with some fixed combinatorics and with vertices of any kind (real, ideal or hyperideal). We find that the supremum is always equal to the volume of the rectification of the 1-skeleton.The theorem is proved by applying a sort of volume-increasing flow to any hyperbolic polyhedron. Singularities may arise in the flow because some strata of the polyhedron may degenerate to lowerdimensional objects; when this occurs, we need to study carefully the combinatorics of the resulting polyhedron and continue with the flow, until eventually we get a rectified polyhedron. Contents 1 Introduction 2 Generalized hyperbolic polyhedra 3 The space of proper polyhedra and the volume function 3.1 The Bao-Bonahon existence and uniqueness theorem for hyperideal polyhedra .

show abstract

Section: Generalized Hyperbolic Polyhedramentioning

confidence: 99%

Section: Generalized Hyperbolic Polyhedramentioning

confidence: 99%

Section: Proof Of Proposition 43mentioning

confidence: 99%

Section: Case 1 Every Limit Edge E ∞mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The maximum volume of hyperbolic polyhedra

Belletti¹

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

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

Belletti

Detcherry

Kalfagianni

et al. 2022

J. Differential Geom.

View full text Add to dashboard Cite

We establish the geometry behind the quantum 6j-symbols under only the admissibility conditions as in the definition of the Turaev-Viro invariants of 3-manifolds. As a classification, we show that the 6-tuples in the quantum 6j-symbols give in a precise way to the dihedral angles of (1) a spherical tetrahedron, (2) a generalized Euclidean tetrahedron, (3) a generalized hyperbolic tetrahedron or (4) in the degenerate case the angles between four oriented straight lines in the Euclidean plane. We also show that for a large proportion of the cases, the 6-tuples always give the dihedral angles of a generalized hyperbolic tetrahedron and the exponential growth rate of the corresponding quantum 6j-symbols equals the suitably defined volume of this generalized hyperbolic tetrahedron. It is worth mentioning that the volume of a generalized hyperbolic tetrahedron can be negative, hence the corresponding sequence of the quantum 6j-symbols could decay exponentially. This is a phenomenon that has never been aware of before.

show abstract

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

Kumar

2020

Preprint

View full text Add to dashboard Cite

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.

show abstract

A maximum volume conjecture for hyperbolic polyhedra

Cited by 5 publications

References 12 publications

The maximum volume of hyperbolic polyhedra

The maximum volume of hyperbolic polyhedra

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

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

Contact Info

Product

Resources

About