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Effective Computation of the Heegaard Genus of 3-Manifolds

Benjamin A. Burton, Finn Thompson

Abstract

The Heegaard genus is a fundamental invariant of 3-manifolds. However, computing the Heegaard genus of a triangulated 3-manifold is NP-hard, and while algorithms exist, little work has been done in making such an algorithm efficient and practical for implementation. Current algorithms use almost normal surfaces, which are an extension of the algorithm-friendly normal surface theory but which add considerable complexity for both running time and implementation. Here we take a different approach: instead of working with almost normal surfaces, we give a general method of modifying the input triangulation that allows us to avoid almost normal surfaces entirely. The cost is just four new tetrahedra, and the benefit is that important surfaces that were once almost normal can be moved to the simpler setting of normal surfaces in the new triangulation. We apply this technique to the computation of Heegaard genus, where we develop algorithms and heuristics that prove successful in practice when applied to a data set of 3,000 closed hyperbolic 3-manifolds; we precisely determine the genus for at least 2,705 of these.

Effective Computation of the Heegaard Genus of 3-Manifolds

Abstract

The Heegaard genus is a fundamental invariant of 3-manifolds. However, computing the Heegaard genus of a triangulated 3-manifold is NP-hard, and while algorithms exist, little work has been done in making such an algorithm efficient and practical for implementation. Current algorithms use almost normal surfaces, which are an extension of the algorithm-friendly normal surface theory but which add considerable complexity for both running time and implementation. Here we take a different approach: instead of working with almost normal surfaces, we give a general method of modifying the input triangulation that allows us to avoid almost normal surfaces entirely. The cost is just four new tetrahedra, and the benefit is that important surfaces that were once almost normal can be moved to the simpler setting of normal surfaces in the new triangulation. We apply this technique to the computation of Heegaard genus, where we develop algorithms and heuristics that prove successful in practice when applied to a data set of 3,000 closed hyperbolic 3-manifolds; we precisely determine the genus for at least 2,705 of these.
Paper Structure (23 sections, 5 theorems, 7 equations, 13 figures, 4 tables)

This paper contains 23 sections, 5 theorems, 7 equations, 13 figures, 4 tables.

Table of Contents

  1. Keywords
  2. Supplementary Material
  3. Acknowledgements
  4. Introduction
  5. Preliminaries
  6. Triangulations
  7. (Almost) Normal Surfaces
  8. Additional Notation
  9. Normalisation Moves
  10. Implementations of Normal and Almost Normal Surfaces
  11. Heegaard Genus
  12. A Gadget for Normalising Almost Normal Surfaces
  13. A General Gadget Construction
  14. Gadget Requirements
  15. Choice of the Gadget
  16. ...and 8 more sections

Key Result

Corollary 3.1

With this construction, for a normal surface $S$ in a triangulation $T$, we can choose a normal surface $S'$ in $T'=T_{\tau}^{abcd(G)}$ where in $G\subset T'$, $S'_G$ has normal coordinates where $\mathbf{S_\tau}=(t_0,t_1,t_2,t_3,q_{01/23},q_{02/13},q_{03/12})$ are the normal coordinates of $S$ in $\tau$. Hence, $S$ is isotopic to $S'$.

Figures (13)

  • Figure 1: Canonical numbering of the vertices of a tetrahedron.
  • Figure 2: The four triangle pieces, and one of the three quadrilateral pieces.
  • Figure 3: Around each intersection $\gamma$, the boundary of a neighbourhood (green) of $\gamma$ is partitioned between the two sheets (red, blue) in one of two ways.
  • Figure 4: A regular exchange is performed on two intersecting pieces in two different ways.
  • Figure 5: Annuli between parallel triangles, non-parallel triangles, a triangle and a quadrilateral, and parallel quadrilaterals; an octagon piece.
  • ...and 8 more figures

Theorems & Definitions (9)

  • Corollary 3.1
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.6
  • proof
  • Corollary 3.7
  • proof