An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary

TL;DR

This paper introduces an algorithm to compute the Heegaard genus of simple 3-manifolds with boundary and to classify all Heegaard splittings of a given genus, enabling the algorithmic calculation of the tunnel number of hyperbolic links.

Contribution

It presents a novel algorithmic approach leveraging almost normal surfaces and a new triangulation structure to determine Heegaard genus and splittings.

Findings

Algorithm successfully computes Heegaard genus for simple 3-manifolds.

Can classify all Heegaard splittings of a specified genus.

Enables algorithmic calculation of tunnel numbers for hyperbolic links.

Abstract

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation.