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Flat fully augmented links are determined by their complements

Christian Millichap, Rolland Trapp

Abstract

In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in $\mathbb{S}^{3}$. This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link complements and behave under homeomorphism. One consequence of this analysis is a complete classification of flat fully augmented link complements that admit multiple reflection surfaces. In addition, our work classifies those symmetries of flat fully augmented link complements which are not induced by symmetries of the corresponding link.

Flat fully augmented links are determined by their complements

Abstract

In this paper, we show that two flat fully augmented links with homeomorphic complements must be equivalent as links in . This requires a careful analysis of how totally geodesic surfaces and cusps intersect in these link complements and behave under homeomorphism. One consequence of this analysis is a complete classification of flat fully augmented link complements that admit multiple reflection surfaces. In addition, our work classifies those symmetries of flat fully augmented link complements which are not induced by symmetries of the corresponding link.
Paper Structure (12 sections, 37 theorems, 13 equations, 22 figures)

This paper contains 12 sections, 37 theorems, 13 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Totally Geodesic Surfaces and Cusps in Flat FALs
  3. Flat FAL Complements with multiple reflection surfaces
  4. Three Reflection Surfaces
  5. Two Reflection Surfaces
  6. Transition to the unique reflection surface case
  7. Signature links and full-swap homeomorphisms
  8. Separating Sets
  9. Complements determine flat FALs
  10. Signature Sublinks
  11. The Standard Ball
  12. Hyperbolicity Conditions

Key Result

Theorem 1.1

Let $\mathcal{A}$ and $\mathcal{A}'$ be flat FALs. Then $(\mathbb{S}^{3}, \mathcal{A})$ is isotopic to $(\mathbb{S}^{3}, \mathcal{A}')$ if and only if $\mathbb{S}^{3} \setminus \mathcal{A}$ is homeomorphic to $\mathbb{S}^{3} \setminus \mathcal{A}'$.

Figures (22)

  • Figure 1: Distinct twisted FALs with homeomorphic complements
  • Figure 2: On the left is a diagram of a link $L$ with three twist regions. The middle diagram shows the corresponding FAL $\mathcal{F}$ obtained from fully augmenting $L$. The right diagram shows the corresponding flat FAL $\mathcal{A}$. Crossing circles of $\mathcal{A}$ are labeled by $c_i$, for $i=1,2,3$.
  • Figure 3: A thrice-punctured sphere with separating geodesics labeled $x,y,z$ and nonseparating geodesics labeled $a,b,c$.
  • Figure 4: Types of non-reflection, thrice-punctured spheres
  • Figure 5: Three reflection surfaces in the Borromean rings complement
  • ...and 17 more figures

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Theorem 2.7
  • ...and 71 more