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Classification of genus-two surfaces in $S^3$

Filippo Baroni

TL;DR

This work provides an algorithmic solution to deciding isotopy of genus-two oriented surfaces in $S^3$ by embedding the problem in the framework of 3-manifold topology. It builds a comprehensive, computable pipeline: (i) decompose the ambient 3-manifold via JSJ into Seifert, $I$-bundle, and strongly simple pieces; (ii) compute mapping class groups and extension data for each piece using a suite of algorithms for Seifert-fibered, i-bundle, and simple components; and (iii) solve a new exponential-equation problem in free groups to control Dehn-twist extensions to handlebodies. A core technical novelty is an algorithmic solution to a free-group problem (via band systems) that underpins the ability to decide when boundary Dehn twists extend across pieces, enabling a finite, constructive decision procedure for genus-two surface isotopy in $S^3$. The results thus unify algorithmic 3-manifold topology with combinatorial group theory to yield a concrete, practically usable isotopy classifier. Overall, the paper advances the computational aspects of 3-manifold topology and provides a concrete tool for genus-two surface classification in $S^3$.

Abstract

We describe an algorithm to decide whether two genus-two surfaces embedded in the 3-sphere are isotopic or not. The algorithm employs well-known techniques in 3-manifolds topology, as well as a new algorithmic solution to a problem on free groups.

Classification of genus-two surfaces in $S^3$

TL;DR

This work provides an algorithmic solution to deciding isotopy of genus-two oriented surfaces in by embedding the problem in the framework of 3-manifold topology. It builds a comprehensive, computable pipeline: (i) decompose the ambient 3-manifold via JSJ into Seifert, -bundle, and strongly simple pieces; (ii) compute mapping class groups and extension data for each piece using a suite of algorithms for Seifert-fibered, i-bundle, and simple components; and (iii) solve a new exponential-equation problem in free groups to control Dehn-twist extensions to handlebodies. A core technical novelty is an algorithmic solution to a free-group problem (via band systems) that underpins the ability to decide when boundary Dehn twists extend across pieces, enabling a finite, constructive decision procedure for genus-two surface isotopy in . The results thus unify algorithmic 3-manifold topology with combinatorial group theory to yield a concrete, practically usable isotopy classifier. Overall, the paper advances the computational aspects of 3-manifold topology and provides a concrete tool for genus-two surface classification in .

Abstract

We describe an algorithm to decide whether two genus-two surfaces embedded in the 3-sphere are isotopic or not. The algorithm employs well-known techniques in 3-manifolds topology, as well as a new algorithmic solution to a problem on free groups.
Paper Structure (4 sections, 5 theorems, 3 equations, 1 figure)

This paper contains 4 sections, 5 theorems, 3 equations, 1 figure.

Table of Contents

  1. Homeomorphisms of 3-manifolds
  2. Rationale
  3. JSJ decomposition
  4. Seifert pieces

Key Result

theorem 1.1

Let $(M,R)$ be an irreducible sufficiently large $3$/̄manifold pair. Then $(M,R)$ admits a JSJ system, which is unique up to isotopy in $M$ fixing $\partial M\mathbin{\begin{tikzpicture}{\draw[line width=0.35pt,line cap=round] (3pt,0) -- (0,6pt);}\end{tikzpicture}}\operatorname{int}(R)$.

Figures (1)

  • Figure 1.1: Local models for a $2$/̄complex defining a Seifert fibration for a $3$/̄manifold $M$. The $2$/̄complex is coloured green, while the fibres are highlighted in red.

Theorems & Definitions (15)

  • definition 1.1
  • definition 1.2
  • definition 1.3
  • definition 1.4
  • definition 1.5
  • theorem 1.1: jaco-shalen-jsj
  • lemma 1.2
  • proof
  • proposition 1.3
  • proof
  • ...and 5 more