Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Practical Software for Triangulating and Simplifying 4-Manifolds

Rhuaidi Antonio Burke

TL;DR

The paper tackles computational understanding of smooth 4-manifolds by developing a pipeline that converts Kirby diagrams into 5-coloured gems and Regina triangulations (via DGT) and introducing Up-Down Simplification (UDS) to shrink triangulations without changing topology. This enables the construction of smaller, more analyzable triangulations of exotic pairs, corks, and plugs, including the smallest known triangulation of the K3 surface (54 pentachora). The work demonstrates concrete results that reveal structural patterns in 4-manifold triangulations and lays groundwork for future exploration of closed exotic manifolds and SPC4-related questions. These developments provide a practical framework for combinatorial and computational investigations into exotic smooth structures and their topological underpinnings.

Abstract

Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.

Practical Software for Triangulating and Simplifying 4-Manifolds

TL;DR

The paper tackles computational understanding of smooth 4-manifolds by developing a pipeline that converts Kirby diagrams into 5-coloured gems and Regina triangulations (via DGT) and introducing Up-Down Simplification (UDS) to shrink triangulations without changing topology. This enables the construction of smaller, more analyzable triangulations of exotic pairs, corks, and plugs, including the smallest known triangulation of the K3 surface (54 pentachora). The work demonstrates concrete results that reveal structural patterns in 4-manifold triangulations and lays groundwork for future exploration of closed exotic manifolds and SPC4-related questions. These developments provide a practical framework for combinatorial and computational investigations into exotic smooth structures and their topological underpinnings.

Abstract

Dimension 4 is the first dimension in which exotic smooth manifold pairs appear -- manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and triangulated 4-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth 4-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of 4-manifolds from Kirby diagrams, as well as a new heuristic for simplifying 4-manifold triangulations. Using these tools, we present new triangulations of several bounded exotic pairs, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the fundamental K3 surface. The small size of these triangulations benefit us by revealing fine structural features in 4-manifold triangulations.
Paper Structure (30 sections, 2 theorems, 3 equations, 29 figures, 1 table)

This paper contains 30 sections, 2 theorems, 3 equations, 29 figures, 1 table.

Table of Contents

  1. Keywords
  2. Acknowledgements
  3. Introduction
  4. Preliminaries
  5. Links and 4-Manifolds
  6. Gems & Crystallisations
  7. Diagrams to Graphs and Triangulations
  8. Construction Details
  9. Up-Down Simplification
  10. The Heuristic
  11. Experimental Results
  12. The $K3$ Surface
  13. Exotic 4-Manifolds
  14. Corks and Plugs
  15. Work Towards Closed Examples
  16. ...and 15 more sections

Key Result

Theorem 3

Every compact $n$-manifold admits an $(n+1)$-coloured graph representing it.

Figures (29)

  • Figure 1: Cartoons illustrating adding a 1-handle versus adding a 2-handle.
  • Figure 2: A framed knot.
  • Figure 3: An example of a Kirby diagram.
  • Figure 4: Constructing a triangulation from a coloured graph.
  • Figure 5: Snappy's PLink Editor
  • ...and 24 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 3: CasaliCristoforiGrasselli2018
  • Conjecture 1
  • Remark 4
  • Definition 5
  • Theorem 6: CorkDecomp-CFHSCorkDecomp-Mat