Studies of Knots, Links, and Other Spatial Configurations

点击翻译按钮获取中文摘要

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 topicsand related research projects are proposed. They range from the knotclassfication problem, a folding problem of polygonal simple curves in the3-space, a possible statistic pattern of the roots of the Jonespolynomial, some problems about representations of braid groups, and thestudy of higher order knot signatures. Some of the problems addressed inthis proposal are major open problems in the field of low dimensionaltopology. The folding problem of polygonal simple spatial curves addressedin this proposal aims at showing that there is essentially no topologicalobstruction for the protein folding problem in molecular biology.The phenomena of knotting and braiding are fundamental in 3-dimensionalspatial structure. A mathematical description and understanding of suchphenomena is essential to our knowledge of the natural world, and is themajor goal of the study of low dimensional topology. The discrete methodsof analyzing 3-dimensional spatial structures developed in the study oflow dimensional topology are proven to be significant to many otherbranches of science and applied research, particularly physics, molecularbiology, and computer graphics. In this proposal, a major effort will bespent on solving an abstract version of the protein folding problem, whichis a central problem in modern molecular biology, using these methods.These methods are also fundamental in the development of a student'sability to visualize and analyze 3-dimensional configurations.