Heegaard Splittings and the Combinatorics of Three-Manifolds

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The PI seeks to understand the algorithmic structure of three-manifolds.To be precise: the PI will work on the Heegaard genus problem and willattack the question of computing the Hempel distance of a Heegaarddiagram. Both questions are difficult due to their connections to thePoincare Conjecture and the three-sphere recognition problem. There areseveral existing tools which spring to mind: on the one hand Haken andRubinstein's combinatorial theory of normal and almost normal surfaceswhile on the other hand there are ideas from the theory of Kleiniangroups, involving the curve complex.To give the flavor of the algorithmic problems involved consider thefollowing "toy" version: your garden hose, lying on the lawn, is verytangled. You must untangle it before putting it away. Is there amechanical procedure (to program into your robot butler) which willuntangle the hose regardless of its starting position? How fast is theprocedure? If the neighbor's child screws the ends of the hose togethercan your butler at least manage to straighten the hose into a circle?Can the butler finish faster if the hose is shorter? How much faster?