Heegaard Splittings and the Combinatorics of Three-Manifolds

Project Details


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?

Effective start/end date6/15/055/31/09


  • National Science Foundation: $92,751.00


Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.