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Proof of Volume Conjecture for twist knots

Sukuse Abe

Abstract

We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.

Proof of Volume Conjecture for twist knots

Abstract

We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.

Paper Structure

This paper contains 5 sections, 18 theorems, 118 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lemmas necessary for Proof of Volume Conjecture for twist knots
  3. First reduction
  4. Second reduction
  5. Proof of the volume conjecture for twist knots

Key Result

Theorem 1.2

Let $K$ be a twist knot, $J_N(K,q)$ be the colored Jones polynomial of $K$, ${\rm Vol}(K)$ be the hyperbolic volume of $S^3 \setminus K$. Then the above equality (50) holds.

Figures (1)

  • Figure 1: Here $2p \, \, (p\in \Z \setminus \{0,1\})$ denote the numbers of half twists in each box.

Theorems & Definitions (35)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1: 11
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 25 more