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

TL;DR

This paper presents an algorithm to determine whether two genus-two surfaces embedded in the 3-sphere are isotopic, utilizing 3-manifold topology techniques and a novel solution to a free group problem.

Contribution

It introduces a new algorithmic approach combining 3-manifold topology and free group problem-solving to classify genus-two surfaces in the 3-sphere.

Findings

Algorithm successfully decides isotopy of genus-two surfaces

Integrates known 3-manifold techniques with a new free group algorithm

Advances computational topology methods for surface classification

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.