Studies of Knots, Links, and Other Spatial Configurations

这个建议的主题是研究拓扑学，几何学，组合学和统计学等空间结构，如结，链接，辫子，多边形简单的空间曲线。提出了五个具体的主题和相关的研究项目。 它们包括纽结的分类问题、3-空间中多边形简单曲线的折叠问题、Jones多项式根的可能统计模式、辫子群表示的一些问题以及高阶纽结签名的研究。在这个建议中解决的一些问题是低维拓扑学领域的主要开放问题。本文提出的多边形简单空间曲线的折叠问题旨在说明分子生物学中蛋白质折叠问题基本上不存在拓扑障碍，打结和编织现象是三维空间结构的基本现象。对这种现象的数学描述和理解对于我们认识自然世界是必不可少的，也是低维拓扑学研究的主要目标。在低维拓扑学研究中发展起来的分析三维空间结构的离散方法，对许多其他科学分支和应用研究，特别是物理学、分子生物学和计算机图形学，都具有重要意义。在这项计划中，主要的努力将是利用这些方法解决蛋白质折叠问题的抽象版本，这是现代分子生物学的核心问题。这些方法也是培养学生可视化和分析三维构型能力的基础。

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.