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

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.

23 pages, 14 figures

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• 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold
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• Counting essential surfaces in 3-manifolds
• Strong Haken via Thin Position
• The Morse-Novikov number of knots under connected sum and cabling
• The tunnel number of all 11 and 12 crossing alternating knots