Gems, Computers and Attractors for 3-Manifolds

The following sections are included:

• Preface

• Contents

The following sections are included:

• Motivations and Objectives

• Attractor at Work: the Case of Poincaré's Homology Sphere S³/〈5,3,2〉

• Translating from Other Presentations into 3-Gems

• The Elements of the Simplifying Dynamics
  • 1-Dipole Cancellation, ρ-Pair Switching and ρ-Move
  • The TS-Configurations and the TS-Moves
  • The U-Move

• Main Theorem and Main Conjectures

The following sections are included:

• Surfaces from (2 + 1)-Graphs

• 3-Manifolds from 3-Gems

• Constructing 3-Gems from Facial Identifications

• 3-Gems from 3-Manifolds
  • Arithmetic Characterization of 3-Gems
  • A Basic Lemma
  • The Superattractor for S¹ × S¹ × S¹

The following sections are included:

• Lens Spaces

• A 4-Parametric Family of 3-Manifolds
  • Defining the Family
  • 1-Dipoles
  • Crystallizations in

• 2-Dipoles and ρ₂-Pairs
  • Dipoles in the Dual Construction
  • The Switching of ρ₂-Pairs

• A Detailed Example of Attraction
  • Cancelling 1-Dipoles
  • Cancelling 2-Dipoles
  • Switching a ρ₂-Pair
  • Final Simplifications to Get

• K₄-Symmetry and Strings Presenting 3-Manifolds
  • Identifying a Special K₄-Symmetry
  • Getting a String Presentation from the σ-Symmetries
  • An Example of Montesinos and Boileau-Zieschang
  • A Conjecture on S¹ x Orientable Surfaces

• A Recoverable Vertex Labelling: The Code
  • DFS-Numbering Algorithms
  • The Code of a Connected Bipartite (n + 1)-Graph
  • The Code of Non-Bipartite Gems

• Heegaard Splittings of 3-Manifolds
  • Arithmetic Properties of (3 + 1)-Graphs
  • Heegaard Splittings

• 3-Gems and π₁, the Fundamental Group
  • Getting 01-Presentation for π₁
  • Combinatorial Explanations
  • Getting a 0-Presentation for π₁

• Framed Links, Blinks and 3-Gems
  • The Smallest Known Hyperbolic 3-Manifold
  • From Blinks to 3-Gems

• From Blinks to Superattractors: Examples
  • The Closed Hyperbolic 3-Manifold:
  • Square-Blink and Tetrahedron-Blink
  • Another Closed Hyperbolic 3-Manifold

• Interesting 3-Manifolds Induced by the Family
  • EUCLID₁ an Euclidean Orientable 3-Manifold
  • The Hyperbolic 3-Manifold
  • The Hyperbolic Dodecahedral Space
  • Homology Tables and Isomorphisms in

The following sections are included:

• A Gem-Based Classification of Surfaces
  • Basic Properties and the Classification Theorem
  • The 2-Dimensional Walking Lemma
  • Special Sequences of 2-Gems Inducing the Closed Surfaces
  • Completeness of the Special Sequences
  • The Classification Theorem

• Handles in Dimension 3
  • The 3-Dimensional Walking Lemma
  • The Superattractors for S¹ × S² and
  • 3-Separation, s¹ × s² and
  • Quartets and Handles
  • ρ₃-Pairs and Handles
  • Double Diagonals and Handles

The following sections are included:

• ρ-Move, TS-Moves and U-Move
  • The ρ-Algorithm
  • The TS-Moves
  • Factoring the TS-Moves into Dipole Moves
  • Clusters of Four Squares
  • The TS_ρ-Algorithm
  • Invariance under the U-Move
  • A -Move on : A Detailed Example
  • The -Algorithm
  • Sketching the Uⁱ-Algorithms

• All Oriented Identifications of a Solid Cube
  • The Orientable Euclidean 3-Manifolds
  • The Lens Spaces
  • Non-Trivial Connected Sums
  • Non-Cyclic Quotients of S³
  • A Remarkably Symmetric 3-Gem

The following sections are included:

• A Catalogue of 3-Manifolds
  • Constructing the Rigid S²-Gems
  • Generation of the Bipartite Rigid 3-gems
  • Codes for Rigid Bipartite 3-Gems

• Understanding Data in the Catalogue
  • Displaying the u⁰-Forests
  • Tables of the ts_ρ*-Function
  • TS_ρ-Classes Ordered by Homology
  • Tabulating the Homology Groups

• The Fundamental Groups of 3-Manifolds |G|, κ(G) ≤ 13
  • An Overview of the Tables
  • Tables of Fundamental Groups
  • Manifolds of Complexity ≤ 10
  • Manifolds of Complexity 11
  • Manifolds of Complexity 12
  • Manifolds of Complexity 13

• Topological Classification
  • Manifolds with Cyclic Finite Fundamental Groups
  • Non-Trivial Connected Sums
  • Manifolds with Handle Number Greater than 0
  • Manifolds with Finite Non-Cyclic Fundamental Groups
  • Covering 3-Gems, Derived 3-Gems
  • Quotients of S³
  • The Euclidean Manifolds
  • The Other 3-Manifolds
  • Summary of the u₀-Uncertainties

The following sections are included:

• Introduction

• Linking Numbers and Self-Linking Sequences

• An Adequate Pair of Dual Decompositions

• The Fundamental Group and a Basis for the Torsion Subgroup

• Getting Loop Words from Paths in a 3-Gem

• Obtaining the Linking Numbers

• Remarks on a Class of 3-Manifolds

The following sections are included:

• A Partition Function
  • Algebraic Ingredients
  • An Example for L_3,1 and the General Case
  • An Example for S³/〈2,2,2〉
  • Distinguishing S³/〈2,2,2〉#L_3,1 and S³/〈2,2,2〉#L_3,2
  • A Brief Explanation for the Invariance

• Tables of Quantum Invariants

The following sections are included:

• Appendices

• Bibliography

• Index