Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots

Huabin Ge, Yunpeng Meng, Chuwen Wang, Yuxuan Yang

Abstract

For a twist knot $\mathcal{K}_{p'}$, let $M$ be the closed $3$-manifold obtained by doing $(p, q)$ Dehn-filling along $\mathcal{K}_{p'}$. In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large $|p| + |q|$ and $|p'|$ for $M$. In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds'' as introduced by Hodgson, Meyerhoff, and Weeks.

On the Volume Conjecture for hyperbolic Dehn-filled 3-manifolds along the twist knots

Abstract

For a twist knot , let be the closed -manifold obtained by doing Dehn-filling along . In this article, we prove that Chen-Yang's volume conjecture holds for sufficiently large and for . In the proof, we construct a new ideal triangulation of the Whitehead link complement which is different from Thurston's triangulation. Our triangulation has led to some new discoveries regarding symmetry, including insights into ``sister manifolds'' as introduced by Hodgson, Meyerhoff, and Weeks.

Paper Structure

This paper contains 30 sections, 66 theorems, 251 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reshetikhin-Turaev invariants
  4. Dilogarithm, Lobachevsky functions and Bloch-Wigner functions
  5. Quantum dilogarithm functions
  6. Geometry of hyperbolic ideal tetrahedron
  7. Dehn filling along twist knots
  8. Continued fractions
  9. Computation of the Reshetikhin-Turaev invariants
  10. Discrete Fourier transform
  11. Potential function $V_r$ and $V$
  12. Poisson summation formula
  13. Symmetry of the Fourier coefficients
  14. Geometry of the critical points
  15. Geometry of Whitehead link complement
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $M=\mathcal{K}_{p'}(p,q)$ be the closed 3-manifold obtained by doing the $(p,q)$ Dehn surgery along the twist knot $\mathcal{K}_{p'}$, then for $|p|+|q|$, $|p'|$ sufficiently large, $r$ positive odd integer, where $C(r)$ is of norm $1$ independent of the geometry structure of $M$, $t(M)$ is a invariant of $\mathcal{K}_{p'}(p,q)$.

Figures (18)

  • Figure 1: A twist knot $\mathcal{K}_{p'}$
  • Figure 2: Whitehead link
  • Figure 3: framed link diagram
  • Figure 5: An ideal tetrahedron in $\mathbb{H}^3$
  • Figure 6: The vertex invariants
  • ...and 13 more figures

Theorems & Definitions (105)

  • Conjecture 1.1: Kashaev-Murakami-Murakami
  • Conjecture 1.2: Chen-Yang
  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Definition 2.1
  • Proposition 2.1: dilog
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemLambda']}
  • Definition 2.2
  • ...and 95 more