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A formula for the volume of two-bridge knots

Julien Marche

Abstract

We give a closed formula for the volume of a two-bridge knot, more precisely for its Bloch invariant. We obtain this formula without triangulating the complement: instead, we derive it from the Hopf formula for the second homology of the fundamental group of the complement and a systematic use of Fox derivatives.

A formula for the volume of two-bridge knots

Abstract

We give a closed formula for the volume of a two-bridge knot, more precisely for its Bloch invariant. We obtain this formula without triangulating the complement: instead, we derive it from the Hopf formula for the second homology of the fundamental group of the complement and a systematic use of Fox derivatives.
Paper Structure (8 sections, 6 theorems, 39 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Commuting meridian and longitude
  3. From the Hopf formula to the bar complex
  4. From the bar complex to the volume
  5. Volume of closed hyperbolic 3-manifolds
  6. Bloch group and Bloch invariant
  7. An explicit formula in the Bloch group
  8. A formula for $\zeta_k(2)$?

Key Result

Theorem 1

Let $0<q<p$ be as above and set $Z_{p,q}=\{x\in \mathbb{C}, P_{p-1}(x)=0\}$. For any $x\in Z_{p,q}$ and $n>0$, we define $z_n=P_n(x)/Q_n(x)$ and set $z_0=\infty$. The volume $V(p,q)$ of $S^3\setminus K(p,q)$ is given by the following formula:

Figures (2)

  • Figure 1: Volumes of two-bridge knots.
  • Figure 2: Conway normal form

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 3
  • Remark 1