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Patterns and Tracks

M. J. Dunwoody

TL;DR

Problem: provide a combinatorial approach to minimal surface theory in triangulated $3$-manifolds and to the $3$-sphere recognition problem via patterns and tracks. Approach: define patterns, tracks, and patterned surfaces; prove that for a triangulated $2$-sphere a maximal non-parallel pattern yields a tree $D_P$ and that an invariant count satisfies $e_P = v_P - 1$, independent of the pattern, with a concrete tetrahedron example. Extend to $3$-manifolds: maximal normal patterns yield a tree $D_P$ whose vertex degrees constrain components to be punctured $3$-balls, and almost normal spheres along with isotopies underpin the Thompson lemma in the Recognition Algorithm. Contributions: a self-contained combinatorial framework for normal and almost normal surface theory with algorithmic implications for recognizing $3$-spheres and studying closed simply connected $3$-manifolds.

Abstract

Patterns in triangulated $2$-spheres and $3$-spheres are investigated. A new proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for $3$-spheres is obtained.

Patterns and Tracks

TL;DR

Problem: provide a combinatorial approach to minimal surface theory in triangulated -manifolds and to the -sphere recognition problem via patterns and tracks. Approach: define patterns, tracks, and patterned surfaces; prove that for a triangulated -sphere a maximal non-parallel pattern yields a tree and that an invariant count satisfies , independent of the pattern, with a concrete tetrahedron example. Extend to -manifolds: maximal normal patterns yield a tree whose vertex degrees constrain components to be punctured -balls, and almost normal spheres along with isotopies underpin the Thompson lemma in the Recognition Algorithm. Contributions: a self-contained combinatorial framework for normal and almost normal surface theory with algorithmic implications for recognizing -spheres and studying closed simply connected -manifolds.

Abstract

Patterns in triangulated -spheres and -spheres are investigated. A new proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for -spheres is obtained.
Paper Structure (4 sections, 2 theorems, 7 figures)

This paper contains 4 sections, 2 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. Patterns and Tracks
  3. Patterns in $2$-spheres
  4. Patterns in $3$-manifolds

Key Result

Theorem 3.1

Every vertex in the tree $D_P$ has degree (valency) one or three. A component of degree one contains one vertex. A component of degree three contains no vertex and is a disc with two smaller discs removed.

Figures (7)

  • Figure 1: Normal Pattern
  • Figure 2: 2
  • Figure 3: 2
  • Figure 4: 2
  • Figure 5: 2
  • ...and 2 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof