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On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot

Qingtao Chen, Shengmao Zhu

Abstract

This is the third article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki and Yokota, we obtain an asymptotic expansion formula for the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral $q$-surgery along the twist knots $\mathcal{K}_p$ at the root of unity $e^{\frac{4π\sqrt{-1}}{r}}$ ($r$ is odd).

On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot

Abstract

This is the third article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki and Yokota, we obtain an asymptotic expansion formula for the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral -surgery along the twist knots at the root of unity ( is odd).

Paper Structure

This paper contains 24 sections, 46 theorems, 359 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Defintion of the Reshetikhin-Turaev invariants
  4. Dilogarithm and Lobachevsky functions
  5. Quantum dilogrithm functions
  6. Saddle point method
  7. Computations of the potential function
  8. Poisson summation formula
  9. Asymptotic expansions
  10. Preparations
  11. The case $m_2\neq 0$
  12. The case $m_2=0$ but $(m_1,m_3)\neq (0,0)$
  13. The contribution of $\hat{h}_{N}(0,0,0)$
  14. Final proof
  15. Appendices
  16. ...and 9 more sections

Key Result

Theorem 1.2

For $(p,q)\in S$, the asymptotic expansion of Reshetikhin-Turaev $RT_{r}(M_{p,q})$ is given by the following form for $d\geq 1$, where $\omega(p,q)$ is given by formula (formula-omegapq) and $\kappa_i(p,q)$ are constants determined by the 3-manifold $M_{p,q}$.

Figures (7)

  • Figure 2.1: The core curve $z$
  • Figure 3.1: Twist knot $\mathcal{K}_p$
  • Figure 4.1: The region $D_0$ lies in the tetrahedron FEBC
  • Figure 5.1: The cube MCB
  • Figure 5.2: The cube $SPM_1$
  • ...and 2 more figures

Theorems & Definitions (76)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 66 more