Efficient Triangulations, Decision Problems & Algorithms

Oklahoma State University, Stillwater OK

Investigators

Abstract

The principal thrust of this research project is further development of efficient triangulations and their applications to the study and understanding of 3-manifolds. 0- and 1-efficient triangulations have lead to a number of new methods and results on triangulations of 3-manifolds, decision problems, algorithms, computational complexity, Heegaard splittings, and Dehn fillings. We propose to develop further our understanding of 0- and 1-efficient triangulations and to form a notion of g-efficient triangulations (g > 1). Our goals are to make an explicit connection between efficient triangulations and geometric structures on 3-manifolds, to use efficient triangulations for a better understanding of Heegaard splittings of 3-manifolds, and to use our methods that reduce a given triangulation to an efficient triangulation to achieve a simplification to the solution of the Homeomorphism Problem for 3-manifolds (Classification of 3-manifolds). In particular, we propose new algorithms for deciding if a 3-manifold is a Haken manifold, for the JSJ decomposition of a 3-manifold and for the recognition of Haken manifolds. Three-dimensional manifolds are mathematical objects which are locally modeled on familiar three-dimensional space. The major problem in the study of three-manifolds is their classification, which is to make a complete list of all three-manifolds without duplications. The study and understanding of three-manifolds toward such a classification is the main objective of this project. In particular, we know that three-manifolds can be considered as a union of building blocks fitting together in a very organized way. For example, tetrahedra may be used as the building blocks; in this case, the collection of tetrahedra and the information about how they fit together is called a triangulation of the three-manifold. All three-manifolds can be triangulated. Thus one of the major strategies toward solving the classification problem is to find methods to recognize a particular three-manifold, having been given the three-manifold in one of its many possible triangulations. Applications of topology of three-manifolds range from questions of protein knotting and unknotting in DNA to the issue of the shape of space (the universe). In particular, the latter may very well come to an issue of recognizing the three-manifold that is our universe. Thus the classification and understanding of three-manifolds has far reaching applications and implications.