Efficient Triangulations and Normal Surface Theory

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Proposal: DMS-9971719PI: William JacoAbstract. This project involves the development and use of efficienttriangulations and normal surface theory in the study and understanding of3-manifolds. A triangulation of a 3-manifold determines a class ofsurfaces in the 3-manifold called normal surfaces. A fundamental principalis that if a manifold contains an interesting surface, then for anytriangulation the manifold contains an interesting normal surface. Theguiding principle of efficient triangulations is reducing unnecessarynormal surfaces and improving the efficiency of computations fortriangulation based algorithms. They have exhibited remarkable success indoing this and, in addition, have revealed new combinatorial informationintimately related to the topology of 3-manifolds. These methods have ledto new, effective algorithms for recognizing the 3-sphere, determining ifa 3-manifold is a Haken manifold, and solving the unknotting problem.Low-dimensional manifolds provide the geometric models of most physicalphenomena. It is natural to study and understand all possible such models.Triangulations and normal surface theory provide an environment for such astudy. Research in these areas is making significant contribution tocomputational topology and complexity theory. It seems possible that thesetechniques will contribute to better computer visualization and possiblyto advances in medical modeling.