We provide an algorithm to determine the Heegaard genus of simple –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.
"An algorithm to determine the Heegaard genus of simple $3$–manifolds with nonempty boundary." Algebr. Geom. Topol. 8 (2) 911 - 934, 2008. https://doi.org/10.2140/agt.2008.8.911