Research in Knots, Links and 3-Manifolds

AbstractAward：DMS-0102231首席研究员：Xiao-Song林特本项目的中心主题是尽可能彻底地探索Jones-Witten和Vassiliev不变量对作为拓扑结构的纽结、链环和三维流形的意义。更具体地说，我们将利用各种概率模型来研究有色Jones多项式的极限;探索我们所发现的Casson不变量的一个简单公式;通过模群的同余子群来理解同余关系Ohtsuki的不变量;寻找从一个纽结补到另一个纽结补的一次映射的标准形;研究纽结复形的上同调;探讨3-流形拓扑学中的经典方法在Vassiliev不变量研究中的适用性;并继续研究琼斯多项式的值和根分布。打结现象是我们生活的空间的一个基本特征，纽结理论是人类科学知识的重要组成部分，因为它旨在理解数学公式的相互作用，空间结构。难怪纽结理论的概念和工具已经被用于数学的许多领域，以及化学，生物学，物理学和计算机科学。例如，遗传学家利用结理论来理解DNA复制的过程和“解开”DNA链的酶的功能，化学家利用结理论来理解和区分不同类型的分子。我们辨别不同结的能力很可能是对我们对空间结构的科学理解的一种考验。

AbstractAward: DMS-0102231Principal Investigator: Xiao-Song LinThe central theme of this project is to explore as thoroughly aspossible the significance of the Jones-Witten and Vassilievinvariants to knots, links, and 3-manifolds as topologicalstructures. To be more specific, we will study the thermodynamiclimit of the colored Jones polynomial using various probabilisticmodels; to explore beyond a simple formulation of the Cassoninvariant we found; to understand the congruence relation amongOhtsuki's invariants through congruence subgroups of the modulargroup; to find the normal form of a degree 1 map from a knotcomplement to another knot complement; to study the cohomology ofknot complexes; to explore the applicability of classicaltechniques in 3-manifold topology to the study of Vassilievinvariants; and to continue the study of value and rootdistributions of the Jones polynomial.The phenomenon of knotting is a fundamental feature of the spacethat we live in, and knot theory is thus an important part ofmankind's scientific knowledge because it is aimed atunderstanding the interplay of mathematical formulae and spacestructures. It is no wonder that concepts and tools originatedfrom knot theory have been used in many areas of mathematics, aswell as in chemistry, biology, physics, and computer science. Forexample, geneticists utilize knot theory to understand theprocesses of DNA replication and the function of enzymes that"unknot" DNA strands, and chemists use knot theory to understandand distinguish between different types of molecules. Our abilityto discern different knots could well be served as a test of ourscientific understanding of space structures.