Studies of Knots, Links, and Other Spatial Configurations

该提案的主题是研究节点、链节、辫子和多边形简单空间曲线等空间构形的拓扑、几何、组合学和统计学。提出了五个具体的课题和相关的研究课题。这些问题包括：纽结分类问题，三维空间中多边形简单曲线的折叠问题，琼斯多项式根的一种可能的统计模式，一些关于辫子群表示的问题，以及高阶纽结签名的研究。该方案中涉及的一些问题是低维拓扑学领域的主要公开问题。本文所讨论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的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中的基本现象。对这种现象的数学描述和理解对于我们对自然界的认识是必不可少的，也是低维拓扑学研究的主要目标。在低维拓扑研究中发展起来的分析三维空间结构的离散方法被证明对许多其他科学和应用研究分支，特别是物理、分子生物学和计算机图形学具有重要意义。在这个提案中，主要的努力将是使用这些方法来解决蛋白质折叠问题的抽象版本，这是现代分子生物学中的一个中心问题。这些方法也是培养学生可视化和分析三维构型的能力的基础。

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.