An algorithm to determine the Heegaard genus of a 3-manifold

TL;DR

This paper presents an algorithmic approach to determine the Heegaard genus of closed, orientable, irreducible, and atoroidal 3-manifolds, establishing finiteness of splittings and providing a practical computational method.

Contribution

It introduces an explicit algorithm to compute the Heegaard genus of atoroidal 3-manifolds, advancing the computational topology of 3-manifolds.

Findings

Finiteness of Heegaard splittings in each genus for the specified class of manifolds

An explicit algorithm to determine the Heegaard genus

Proof of the algorithm's correctness and applicability

Abstract

We give an algorithmic proof of the theorem that a closed orientable irreducible and atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. The proof gives an algorithm to determine the Heegaard genus of an atoroidal 3-manifold.