Heegaard Splittings and the Combinatorics of Three-Manifolds

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The PI seeks to understand the algorithmic structure of three-manifolds. To be precise: the PI will work on the Heegaard genus problem and will attack the question of computing the Hempel distance of a Heegaard diagram. Both questions are difficult due to their connections to the Poincare Conjecture and the three-sphere recognition problem. There are several existing tools which spring to mind: on the one hand Haken and Rubinstein's combinatorial theory of normal and almost normal surfaces while on the other hand there are ideas from the theory of Kleinian groups, involving the curve complex. To give the flavor of the algorithmic problems involved consider the following "toy" version: your garden hose, lying on the lawn, is very tangled. You must untangle it before putting it away. Is there a mechanical procedure (to program into your robot butler) which will untangle the hose regardless of its starting position? How fast is the procedure? If the neighbor's child screws the ends of the hose together can 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?