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Hyperbolicity of Staked Links and Lower Bounds on Their Volumes

Colin Adams, Francisco Gomez-Paz, Jiachen Kang, Lukas Krause

Abstract

We define a class of links in handlebodies called ``charm bracelets," which are a subset of staked links. We provide tools to construct infinitely many such hyperbolic links and bound the corresponding volumes from below in terms of volumes corresponding to the individual charms.

Hyperbolicity of Staked Links and Lower Bounds on Their Volumes

Abstract

We define a class of links in handlebodies called ``charm bracelets," which are a subset of staked links. We provide tools to construct infinitely many such hyperbolic links and bound the corresponding volumes from below in terms of volumes corresponding to the individual charms.
Paper Structure (11 sections, 25 theorems, 1 equation, 21 figures)

This paper contains 11 sections, 25 theorems, 1 equation, 21 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolicity of charm bracelets
  4. Results
  5. Notation
  6. Outline of the proof of the General Theorem
  7. Sheets and Essentiality
  8. Eliminating Essential Spheres and Disks
  9. Eliminating Essential Tori
  10. Eliminating Essential Annuli
  11. Proof of Theorem

Key Result

Theorem 1.1

If each charm in an even charm bracelet $B = (C_1, C_2, \dots , C_{2m})$ is $2m$-hyperbolic, then $B$ is hyperbolic and

Figures (21)

  • Figure 1: Realizing a staked knot as a knot in a handlebody.
  • Figure 2: A charm $C$ from a charm bracelet.
  • Figure 3: An example of the application of Theorem 1.1.
  • Figure 4: Examples of bongles.
  • Figure 5: Concatenating two $n$-stranded charms.
  • ...and 16 more figures

Theorems & Definitions (64)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 54 more