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Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture

Shashini Marasinghe

TL;DR

The paper addresses the large-$r$ behavior of $TV_r(M,q)$ for oriented Seifert fibered 3-manifolds and verifies the generalized TV-volume conjecture for broad Seifert families, including manifolds with boundary. It leverages the TV–RT relationship via doubles $D(M)$ and Hansen’s explicit $RT_r$ formulas for Seifert manifolds, performing a subsequence analysis along $r=kA$ with $A= ext{lcm}(a_1, obreak \dots, obreak a_n)$ to show the growth rate $LTV(M)=0$. The results extend volume-conjecture phenomena beyond hyperbolic manifolds to Seifert-fibered classes, with corollaries for Dehn fillings and links of zero simplicial volume. By connecting quantum invariants to classical geometric data in non-hyperbolic 3-manifolds, the work broadens the scope of TV-volume verifications and informs future studies in quantum topology and 3-manifold invariants.

Abstract

We study the large $r$ asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root $q=e^\frac{2πi}{r}$. As an application, we prove the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty and non-empty boundary.

Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture

TL;DR

The paper addresses the large- behavior of for oriented Seifert fibered 3-manifolds and verifies the generalized TV-volume conjecture for broad Seifert families, including manifolds with boundary. It leverages the TV–RT relationship via doubles and Hansen’s explicit formulas for Seifert manifolds, performing a subsequence analysis along with to show the growth rate . The results extend volume-conjecture phenomena beyond hyperbolic manifolds to Seifert-fibered classes, with corollaries for Dehn fillings and links of zero simplicial volume. By connecting quantum invariants to classical geometric data in non-hyperbolic 3-manifolds, the work broadens the scope of TV-volume verifications and informs future studies in quantum topology and 3-manifold invariants.

Abstract

We study the large asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root . As an application, we prove the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty and non-empty boundary.

Paper Structure

This paper contains 11 sections, 11 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Oriented Seifert fibered 3-manifolds
  5. Turaev-Viro invariants
  6. Notation
  7. Witten-Reshetikhin-Turaev invariants
  8. Witten-Reshetikhin-Turaev invariants of oriented Seifert fibered 3-manifolds
  9. Relationship between $SO(3)$-Turaev-Viro invariants and $SU(2)$-Witten-Reshetikhin-Turaev invariants
  10. Proof of Theorem \ref{['main.A']}
  11. Some examples

Key Result

Theorem 1.3

Let $M$ be an oriented Seifert fibered 3-manifold with boundary, described by the symbol Suppose that there is an integer $\gamma > 0$ and $\boldsymbol{\mu}=(\mu_1,\ldots ,\mu_n)$ with $\mu_j=\pm 1$ such that for $j=1,\ldots,n$. Then, $M$ and $D(M)$ satisfy Conjecture con:TV volume. That is, $LTV(M)=LTV(D(M))=0$.

Theorems & Definitions (16)

  • Definition 1.1: Section 1.1 detcherry2020gromov
  • Conjecture 1.2: Turaev-Viro invariants volume conjecture 8.1 detcherry2020gromov
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Definition 2.1: Definition 2.8 detcherry2018turaev
  • Theorem 2.2: kirby19913, blanchet1995topological
  • Lemma 2.3: Theorem 8.4 hansen2001reshetikhin
  • Remark 2.4
  • Theorem 2.5: robertproof
  • ...and 6 more