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Crushing Surfaces of Positive Genus

Benjamin A. Burton, Thiago de Paiva, Alexander He, Connie On Yu Hui

TL;DR

The paper develops a detailed atomic-framework analysis to understand the topological effects of crushing closed normal surfaces of positive genus in 3-manifold triangulations. By extending normalisation and barrier techniques to ideal/invalid vertices, it proves a main theorem: if X is irreducible, ∂-irreducible, anannular, and contains no two-sided Möbius bands, then crushing along a maximal normal surface S yields a triangulation where a component is an ideal triangulation of the cut submanifold X and all other components are 3-spheres, with the submanifold’s triangulation complexity strictly less than that of the ambient manifold. The approach avoids Matveev’s spine machinery by working entirely with triangulations and cell decompositions, enabling practical algorithmic applications via Regina and yielding insight into JSJ decompositions and satellite knot constructions. The results then translate into concrete inequalities for triangulation complexity of submanifolds, including hyperbolic JSJ pieces, satellite knot exteriors, and rod complements in T^3, highlighting the broad impact on computational 3-manifold topology and related geometric questions.

Abstract

The operation of crushing a normal surface has proven to be a powerful tool in computational $3$-manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: $2$-spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions.

Crushing Surfaces of Positive Genus

TL;DR

The paper develops a detailed atomic-framework analysis to understand the topological effects of crushing closed normal surfaces of positive genus in 3-manifold triangulations. By extending normalisation and barrier techniques to ideal/invalid vertices, it proves a main theorem: if X is irreducible, ∂-irreducible, anannular, and contains no two-sided Möbius bands, then crushing along a maximal normal surface S yields a triangulation where a component is an ideal triangulation of the cut submanifold X and all other components are 3-spheres, with the submanifold’s triangulation complexity strictly less than that of the ambient manifold. The approach avoids Matveev’s spine machinery by working entirely with triangulations and cell decompositions, enabling practical algorithmic applications via Regina and yielding insight into JSJ decompositions and satellite knot constructions. The results then translate into concrete inequalities for triangulation complexity of submanifolds, including hyperbolic JSJ pieces, satellite knot exteriors, and rod complements in T^3, highlighting the broad impact on computational 3-manifold topology and related geometric questions.

Abstract

The operation of crushing a normal surface has proven to be a powerful tool in computational -manifold topology, with applications both to triangulation complexity and to algorithms. The main difficulty with crushing is that it can drastically change the topology of a triangulation, so applications to date have been limited to relatively simple surfaces: -spheres, discs, annuli, and closed boundary-parallel surfaces. We give the first detailed analysis of the topological effects of crushing closed essential surfaces of positive genus. To showcase the utility of this new analysis, we use it to prove some results about how triangulation complexity interacts with JSJ decompositions and satellite knots; although similar applications can also be obtained using techniques of Matveev, our approach has the advantage that it avoids the machinery of almost simple spines and handle decompositions.
Paper Structure (23 sections, 17 theorems, 2 equations, 29 figures)

This paper contains 23 sections, 17 theorems, 2 equations, 29 figures.

Table of Contents

  1. Keywords
  2. Introduction
  3. Acknowledgements.
  4. Preliminaries
  5. Triangulations and cell decompositions
  6. Decomposing along curves and surfaces
  7. Normal surfaces
  8. Barriers and normalisation
  9. Crushing via atomic moves
  10. Atomic moves on cell decompositions with ideal vertices
  11. Flattening triangular and bigon pillows
  12. Flattening bigon faces
  13. User guide
  14. Interaction with vertices
  15. The case where ungluing gives a single boundary bigon path
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose that $X$ is irreducible, $\partial$-irreducible and anannular, and that it contains no two-sided properly embedded Möbius bands. Also, suppose $S$ is maximal. Then $\mathcal{T}^\ast$ is a valid triangulation such that:

Figures (29)

  • Figure 1: Two tetrahedra glued together along a single pair of triangular faces.
  • Figure 2: Some examples of the non-tetrahedron cells that we will encounter.
  • Figure 3: Cutting along an embedded closed curve in a surface.
  • Figure 4: The seven types of elementary disc.
  • Figure 5: A portion of a normal surface built entirely out of triangles.
  • ...and 24 more figures

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Remark
  • Remark
  • Theorem 4
  • Lemma 7: Crushing lemma
  • Lemma 8
  • Lemma 10: Flattening triangular pillows
  • Lemma 11: Flattening bigon pillows
  • Claim 2
  • ...and 18 more