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Seifert cobordisms and the Chen-Yang volume conjecture

Renaud Detcherry, Efstratia Kalfagianni, Shashini Marasinghe

TL;DR

This work analyzes the large-$r$ behavior of Turaev-Viro invariants $TV_r(M;q^2)$ for manifolds with toroidal boundary under gluing by Seifert-fibered cobordisms, proving a robust, polynomial-bounded relation between $TV_r$ across such gluings. By leveraging the SO$_3$ TQFT framework, the authors establish invertibility and polynomial growth bounds for RT$_r$ maps associated with Seifert pieces, and extend these bounds to graphs of Seifert pieces (graph manifolds) via plumbed cobordisms. Consequently, they demonstrate that the Turaev-Viro volume conjecture, relating asymptotics of $TV_r$ to the simplicial (Gromov) volume, is preserved under Seifert gluing, enabling verification of the Chen-Yang conjecture for broad families: all Seifert fibered manifolds with boundary and large classes of graph manifolds. The results also discuss hyperbolic cobordisms and how Dehn fillings with zero volume constrain the growth of RT$_r$-inverse norms, highlighting avenues for future exploration of exponential growth phenomena in quantum invariants.

Abstract

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M; e^{\frac{2πi}{r}})$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $\partial M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

Seifert cobordisms and the Chen-Yang volume conjecture

TL;DR

This work analyzes the large- behavior of Turaev-Viro invariants for manifolds with toroidal boundary under gluing by Seifert-fibered cobordisms, proving a robust, polynomial-bounded relation between across such gluings. By leveraging the SO TQFT framework, the authors establish invertibility and polynomial growth bounds for RT maps associated with Seifert pieces, and extend these bounds to graphs of Seifert pieces (graph manifolds) via plumbed cobordisms. Consequently, they demonstrate that the Turaev-Viro volume conjecture, relating asymptotics of to the simplicial (Gromov) volume, is preserved under Seifert gluing, enabling verification of the Chen-Yang conjecture for broad families: all Seifert fibered manifolds with boundary and large classes of graph manifolds. The results also discuss hyperbolic cobordisms and how Dehn fillings with zero volume constrain the growth of RT-inverse norms, highlighting avenues for future exploration of exponential growth phenomena in quantum invariants.

Abstract

We study the large asymptotic behavior of the Turaev-Viro invariants of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of . We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.
Paper Structure (12 sections, 16 theorems, 46 equations)

This paper contains 12 sections, 16 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. TQFT and Turaev-Viro invariants
  3. Preliminaries
  4. Genus one mutation
  5. TQFT maps of Seifert fibered spaces
  6. Gluing Theorems
  7. Seifert cobordisms
  8. Invertible cable spaces
  9. Proof of Theorem \ref{['seifertglue']}
  10. Plumbed Cobordisms
  11. Volume Conjecture applications
  12. Hyperbolic cobordisms

Key Result

Theorem 1.1

Let $S$ be a Seifert fibered 3-manifold with at least two boundary components and let $M$ be any 3-manifold with toroidal boundary. Then, for any 3-manifold $M'$ obtained by gluing $S$ along a component of $\partial S$ to a component of $\partial M$, there exist constants $A$ and $K>0$, and a finite for all odd $r$ not divisible by any $p\in I$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 23 more