Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finding large counterexamples by selectively exploring the Pachner graph

Benjamin A. Burton, Alexander He

TL;DR

Here, it is shown that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations.

Abstract

We often rely on censuses of triangulations to guide our intuition in $3$-manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to $10$ tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain $3$-manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the $3$-manifold.

Finding large counterexamples by selectively exploring the Pachner graph

TL;DR

Here, it is shown that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations.

Abstract

We often rely on censuses of triangulations to guide our intuition in -manifold topology. However, this can lead to misplaced faith in conjectures if the smallest counterexamples are too large to appear in our census. Since the number of triangulations increases super-exponentially with size, there is no way to expand a census beyond relatively small triangulations; the current census only goes up to tetrahedra. Here, we show that it is feasible to search for large and hard-to-find counterexamples by using heuristics to selectively (rather than exhaustively) enumerate triangulations. We use this idea to find counterexamples to three conjectures which ask, for certain -manifolds, whether one-vertex triangulations always have a "distinctive" edge that would allow us to recognise the -manifold.
Paper Structure (31 sections, 14 theorems, 6 equations, 10 figures, 5 algorithms)

This paper contains 31 sections, 14 theorems, 6 equations, 10 figures, 5 algorithms.

Table of Contents

  1. Keywords
  2. Acknowledgements
  3. Introduction
  4. The conjectures
  5. Using a targeted search to find counterexamples
  6. Preliminaries
  7. Triangulations
  8. The 2-3 and 3-2 moves
  9. Embedded surfaces
  10. Normal surfaces
  11. Lens spaces
  12. Seifert fibre spaces
  13. Key tools
  14. Handlebody recognition
  15. Detecting Seifert fibre edges
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be a lens space that is neither $\mathbb{R}P^3$ nor a prism manifoldLackenby and Schleimer use a slightly different definition of prism manifold than the one we give in Section subsec:sfs., and let $\mathcal{T}$ be any triangulation of $\mathcal{M}$. Then the 86 iterated barycentri

Figures (10)

  • Figure 1: Three knots with tunnel number equal to one.
  • Figure 2: Gluing two faces of a single tetrahedron with a "twist" gives a one-vertex triangulation of the solid torus.
  • Figure 3: The 2-3 and 3-2 moves.
  • Figure 4: An inessential compression disc $D$ for a surface $S$.
  • Figure 5: An inessential $\partial$-compression disc $D$ for a surface $S$.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Conjecture 1
  • Theorem : Lackenby and Schleimer LackenbySchleimer2022
  • Conjecture 2
  • Conjecture 3
  • Theorem 4
  • Theorem 6
  • Theorem 7
  • Proposition 8
  • Theorem 10: Correctness
  • Proposition 11
  • ...and 11 more