Efficient Triangulations of Three-Manifolds

Oklahoma State University, Stillwater OK

Investigators

Abstract

DMS-0204707 William H. Jaco This project involves the development and use of efficient triangulations in the study and understanding of 3--manifolds. A triangulation of a 3-manifold determines a class of surfaces in the 3--manifold called normal surfaces. A fundamental observation is that in most cases when a 3--manifold contains an interesting surface, then for any triangulation of the 3--manifold the same interesting feature is exhibited by a normal surface in the triangulation. This has provided a powerful method for studying 3-manifolds and solving certain decision problems. Normal surface theory has, for example, lead to the theoretical resolution of determining when two 3--manifolds, which contain injective surfaces, are homeomorphic, to the classification of these same manifolds, and to the determination of whether a knot in the three-sphere is the unknot. However, even with the presence of interesting normal surfaces, a given triangulation may have numerous uninteresting normal surfaces, which make theoretical arguments quite tedious and computational attempts nearly impossible. The guiding principle of efficient triangulations is reducing unnecessary normal surfaces and improving the efficiency of computations for triangulation based algorithms. They have exhibited remarkable success in doing this. However, more surprisingly, efficient triangulations have exposed interesting relations between one-vertex and ideal triangulations of three-manifolds and new combinatorial structures, which seem to be intimately related to the topology of the three-manifold. This project will explore efficient triangulations with particular emphasis of their application to the Homeomorphism Problem/Classification Problem for three-manifolds and issues of computational complexity. Low-dimensional topology brings together many areas of mathematical research and provides a common ground for interaction and advance across all of mathematics. It provides a natural geometric model of most physical phenomena. Research in this area is making significant contribution to computational geometry and topology and complexity theory. It has consequences in physics, computer visualization and medical and biological modeling.