Invariants and geometry of knots and 3-manifolds

结和 3 流形的不变量和几何

• 批准号: 1105843
• 负责人: Efstratia Kalfagianni
• 金额: $ 19.04万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105843&HistoricalAwards=false
• 关键词: Invariants; geometry; knots; manifolds

The project addresses relations between geometry, topology and combinatorial invariants of 3-manifolds. The PI will establish concrete connections between geometry and topological descriptions, properties, and quantum invariants of links and 3-manifolds. One part of the project will combine several techniques, developed both by the PI and her collaborators and by others, to derive estimates of hyperbolic volumes and other geometric invariants of 3-manifolds from topological data such as Dehn surgery presentations, link diagrams and group actions. A second part will establish relations between the Jones knot polynomials (and the Khovanov homology), essential surfaces, and geometric structures of knot complements. A third part will study skein link theory in 3-manifolds, its invariants, and investigate its interaction with the detailed structures coming from the geometrization picture. The research also involves graduate students currently working with PI.The research of the project lies in the area of 3-dimensional topology. The central objects of this study 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 for 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 3-dimensional topology. The main theme of this project is to establish concrete such connections and explore their ramifications and applications to other areas of mathematics.

该项目解决几何，拓扑和组合不变量的3流形之间的关系。PI将建立几何和拓扑描述、性质以及链路和3流形的量子不变量之间的具体联系。该项目的一部分将结合PI和她的合作者以及其他人开发的几种技术，从拓扑数据（如Dehn手术演示、链接图和群体行为）中推导出双曲体积和3流形的其他几何不变量的估计。第二部分将建立琼斯结多项式（和Khovanov同调）、基本曲面和结补的几何结构之间的关系。第三部分将研究3流形中的绞链理论及其不变量，并研究其与来自几何化图的详细结构的相互作用。该研究还包括目前与PI合作的研究生。本课题的研究方向为三维拓扑领域。这项研究的中心对象是被称为3流形的空间。三维流形是一种局部看起来像普通三维空间但整体结构复杂的物体。三维拓扑学的一个重要部分也是对结点（以某种纠缠方式嵌入在三维流形中的环）及其分类的研究。瑟斯顿几何猜想的解已经确立了3-流形（以及其中的节的补）分解成承认显式几何的碎片，而双曲几何是更经常出现的一种。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几何信息的方法是很自然和重要的。拓扑学家研究3-流形的方法之一是使用不变量。在过去的几十年里，源于物理学的思想使数学家们发现了结和3流形的各种不变量。了解拓扑、组合量和不变量与几何的联系是三维拓扑学的中心和重要目标。该项目的主要主题是建立具体的这种联系，并探索其分支和应用到其他数学领域。