An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary

Abstract

We provide an algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty 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 on a new structure called a partially flat angled ideal triangulation.