Hyperbolic 3-Manifolds as Discretized Configuration Spaces of Simple Graphs Open Access

Abstract

A discretized configuration space is a topological space of possible configurations
of particles in a graph in which multiple particles are not allowed in neighbouring
edges. In this paper, we consider discretized configuration spaces of three particles on
1-dimensional simple graphs, in particular, the discretized configuration spaces D3(K7)
and D3(K4,4). We prove that in both cases, removing certain vertices from the discretized
configuration spaces on three particles results in complete finite-volume hyperbolic
3-manifolds. We describe their construction and triangulations by cubes and
tetrahedra. We also discuss their commensurability class in relation to each other and
to the complement of the figure-8 knot.

Table of Contents

Contents
1 Introduction 1
2 Prerequisites 2
2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Manifolds and Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Cell Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Group Theory and Algebraic Topology . . . . . . . . . . . . . . . . . . . . . 7
3 Preliminaries and Denitions 9
3.1 Motion-Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Formal Denition of Conguration Spaces . . . . . . . . . . . . . . . . . . . 12
3.3 The Premise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Discretized Space of K7 18
4.1 2-dimensional forerunner: D2(K5) . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Geometric Properties of M1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 Other Discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Discretized Space of K4;4 27
5.1 2-dimensional forerunner: D2(K3;3) . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.3 Geometric Properties of M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Other Discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 The Relationship 36

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	3-Manifolds topology hyperbolic geometry
Committee Chair / Thesis Advisor	Abrams, Aaron, Emory University
Committee Members	Hamilton, Emily, Emory University Klumpp, Tilman, Emory University

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