Studies of Knots, Links, and Other Spatial Configurations

University Of California-Riverside, Riverside CA

Investigators

Abstract

The subject of this proposal is to study the topology, geometry, combinatorics, and statistics of such spatial configurations as knots, links, braids, and polygonal simple spatial curves. Five specific topics and related research projects are proposed. They range from the knot classfication problem, a folding problem of polygonal simple curves in the 3-space, a possible statistic pattern of the roots of the Jones polynomial, some problems about representations of braid groups, and the study of higher order knot signatures. Some of the problems addressed in this proposal are major open problems in the field of low dimensional topology. The folding problem of polygonal simple spatial curves addressed in this proposal aims at showing that there is essentially no topological obstruction for the protein folding problem in molecular biology. The phenomena of knotting and braiding are fundamental in 3-dimensional spatial structure. A mathematical description and understanding of such phenomena is essential to our knowledge of the natural world, and is the major goal of the study of low dimensional topology. The discrete methods of analyzing 3-dimensional spatial structures developed in the study of low dimensional topology are proven to be significant to many other branches of science and applied research, particularly physics, molecular biology, and computer graphics. In this proposal, a major effort will be spent on solving an abstract version of the protein folding problem, which is a central problem in modern molecular biology, using these methods. These methods are also fundamental in the development of a student's ability to visualize and analyze 3-dimensional configurations.