arXiv subjects

Giulio Belletti

Publications and source records attributed to Giulio Belletti.

10 recordsLinked to original sources

arXiv2026

Embeddability between 3-manifolds is not stable

We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For instance, as a weak version of our main theorem, if $M$ is a rational homology sphere then for any $k\geq 1$ the $3$-manifold $N'$ can be chosen to be $Y_k$-equivalent to $N.$ Our techniques rely on the construction of $3$-manifolds with complicated Frohman--Kania-Bartoszyńska ideals, using the strong approximation for $\mathrm{SO}_3$-Witten-Reshetikhin-Turaev quantum representations of mapping class groups of surfaces.

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arXiv2025

An equivalent condition for q-holonomicity

We show that a sequence is q-holonomic if and only if it satisfies the elimination property for any subset of variables. The same result also holds for holonomic sequences. As an application, we prove several conjectured closure properties for q-holonomic sequences. We also prove that Jones-style sequences for links in any closed $3$-manifold are q-holonomic, which in turn implies that the Reshetikhin-Turaev invariants are q-holonomic in the colors.

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arXiv2025

An effective proof of finiteness for Kauffman bracket skein modules

We prove a version of the finiteness conjecture for Kauffman bracket skein modules of $3$-manifolds with boundary, which was introduced by the second author in \cite{Det21}. In particular our methods, which are constructive, give an alternative proof of Witten's finiteness conjecture for the Kauffman bracket skein modules of closed $3$-manifolds, which was originally proved in \cite{GJS19}. Moreover, as a corollary we show that the peripheral ideal of any link is non-empty, answering a question of Frohman, Gelca and Lofaro \cite{FGL02}.

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arXiv2024

On torsion in the Kauffman bracket skein module of $3$-manifolds

We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\mathcal{S}(M)$ has torsion as a $\mathbb{Z}[A^{\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\mathcal{S}(\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface.

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arXiv2024

An upper bound conjecture for the Yokota invariant

We conjecture an upper bound on the growth of the Yokota invariant of polyhedral graphs, extending a previous result on the growth of the $6j$-symbol. 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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arXiv2023

Asymptotics of quantum $6j$-symbols and generalized hyperbolic tetrahedra

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.

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arXiv2022

The volume conjecture for polyhedra implies the Stoker conjecture

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result of Montcouquiol and Weiss, saying that dihedral angles are local coordinates for compact polyhedra with angles $\leq π$.

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arXiv2021

Discrete Fourier transforms, quantum $6j$-symbols and deeply truncated tetrahedra

The asymptotic behavior of quantum $6j$-symbols is closely related to the volume of truncated hyperideal tetrahedra\,\cite{C}, and plays a central role in understanding the asymptotics of the Turaev-Viro invariants of $3$-manifolds. In this paper, we propose a conjecture relating the asymptotics of the discrete Fourier transforms of quantum $6j$-symbols on one hand, and the volume of deeply truncated tetrahedra of various types on the other. As supporting evidence, we prove the conjecture in the case that the dihedral angles are sufficiently small, and provide numerical calculations in the case that the dihedral angles are relatively big. A key observation is a relationship between quantum $6j$-symbols and the co-volume function of deeply truncated tetrahedra, which is of interest in its own right. More ambitiously, we extend the conjecture to the discrete Fourier transforms of the Yokota invariants of planar graphs and volume of deeply truncated polyhedra, and provide supporting evidence.

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arXiv2020

The maximum volume of hyperbolic polyhedra

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 lower-dimensional 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.

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arXiv2020

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

We prove the Turaev-Viro invariants volume conjecture for a "universal" class of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev-Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $q=e^{\frac{2πi}{r}}.$

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