Research in Knots, Links and 3-Manifolds

摘要：项目负责人：林晓松。本项目的中心主题是尽可能深入地探讨结点、链路和3流形拓扑结构的Jones-Witten不变量和vassilievinvariant的意义。更具体地说，我们将使用各种概率模型研究彩色琼斯多项式的热力学极限；探索超越我们发现的卡松不变式的简单公式；通过模群的同余子群来理解gohtsuki不变量之间的同余关系；求从一个结补到另一个结补的1度映射的正规形式；研究结配合物的上同性；探索经典的3流形拓扑技术在vassilievinvariant研究中的适用性；并继续研究琼斯多项式的值分布和根分布。打结现象是我们生活的空间的一个基本特征，因此打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和空间结构的相互作用。毫无疑问，起源于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可以用来测试我们对空间结构的科学理解。

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.