Geometric structures and invariants of links and 3-manifolds

链接和 3 流形的几何结构和不变量

• 批准号: 1404754
• 负责人: Efstratia Kalfagianni
• 金额: $ 22.44万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1404754&HistoricalAwards=false
• 关键词: Geometric; structures; invariants; links; manifolds

The research of this project falls in the area of 3-dimensional topology. The central objects of study in this area are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3- dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. The solution of Thurston's Geometrization Conjecture has established that 3-manifolds (and complements of knots in them) decompose into pieces that admit explicit geometries and that hyperbolic geometry is the one that appears more often. In practice, however, 3-manifolds are often given in terms of combinatorial topological descriptions and it is both natural and important to seek ways to deduce geometric information from these descriptions. One of the ways that topologists have been approaching the study of 3- manifolds is through the use of invariants. In the last few decades ideas originated in physics led mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. Understanding the connections of topological and combinatorial quantities and invariants to geometry is a central and important goal of low dimensional topology. The main theme of this project is to establish such connections and explore their ramifications and applications to other areas of mathematics.This project will establish relationships between geometry and combinatorial descriptions, properties, and quantum invariants of links and 3-manifolds. The PI has developed a setting for establishing new unexpected relations between the colored Jones link polynomials, the topology and geometry of essential surfaces in link complements, and hyperbolic geometry. One part of the project will continue developing this theory and exploring its applications. Another part, will combine several new techniques, to develop methods for recognizing geometric structures on 3-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data. A third part will study skein link theory in 3-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds. A fourth part will explore the applicability of quantum knot invariants to classical questions in knot theory and search for a classification of crossing changes that do not alter the topology of the underlying knots. The project also involves the research of graduate students currently working with PI.

该项目的研究属于3维拓扑领域。该领域的中心研究对象是称为三流形的空间。 3 流形是一种局部看起来像普通 3 维空间的对象，但其全局结构可能很复杂。 3 维拓扑的一个重要部分也是对结（以某种纠缠方式嵌入 3 流形中的环）及其分类的研究。瑟斯顿几何化猜想的解决方案已经确定，3-流形（以及其中的结的补集）分解为允许显式几何的片段，并且双曲几何是最常出现的几何。然而，在实践中，3-流形通常以组合拓扑描述的形式给出，寻求从这些描述中推导出几何信息的方法既自然又重要。拓扑学家研究三流形的方法之一是使用不变量。在过去的几十年里，起源于物理学的想法引导数学家发现了各种纽结和 3 流形的不变量。理解拓扑和组合量以及不变量与几何的联系是低维拓扑的核心和重要目标。该项目的主题是建立这种联系并探索它们在数学其他领域的影响和应用。该项目将建立几何与组合描述、属性以及连杆和三流形的量子不变量之间的关系。 PI 开发了一种设置，用于在彩色琼斯链接多项式、链接补体中基本曲面的拓扑和几何以及双曲几何之间建立新的意想不到的关系。该项目的一部分将继续发展这一理论并探索其应用。另一部分将结合几种新技术，开发从纯组合输入中识别 3 流形上的几何结构的方法，并从拓扑数据中得出几何量的估计。第三部分将研究 3-流形中的绞纱连接理论、其不变量及其与 3-流形几何分解的相互作用。 第四部分将探讨量子结不变量对结理论中经典问题的适用性，并寻找不改变底层结拓扑的交叉变化的分类。该项目还涉及目前与 PI 合作的研究生的研究。