Invariants and geometry of knots and 3-manifolds

该项目解决了3-流形的几何、拓扑和组合不变量之间的关系。PI将在几何和拓扑描述、性质以及环和3-流形的量子不变量之间建立具体的联系。项目的一部分将结合由PI和她的合作者以及其他人开发的几种技术，从Dehn手术演示、链接图和群动作等拓扑数据中推导出三维流形的双曲体积和其他几何不变量的估计。第二部分将建立Jones纽结多项式(和Khovanov同调)、本质曲面和纽结补的几何结构之间的关系。第三部分将研究三维流形中的斜链理论及其不变量，并研究它与几何化图像中的细节结构之间的相互作用。这项研究也涉及到目前在PI工作的研究生。该项目的研究领域是三维拓扑学。本研究的中心对象是称为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的对象，但其全局结构可能很复杂。三维拓扑学的一个重要部分也是对纽结(以某种纠缠方式嵌入到三维流形中的环)及其分类的研究。瑟斯顿的几何化猜想的解确立了3-流形(以及其中的纽结的补集)分解成允许显式几何的片断，并且双曲几何出现得更频繁。然而，在实践中，三维流形往往是以组合拓扑描述的形式给出的，因此寻找从这些描述中推导几何信息的方法是很自然也很重要的。拓扑学家一直在研究三维流形的方法之一是通过使用不变量。在过去的几十年里，起源于物理学的想法导致了数学家们发现了各种纽结和三维流形的不变量。理解拓扑量、组合量和不变量与几何的关系是三维拓扑学的中心和重要目标。这个项目的主要主题是建立具体的这种联系，并探索它们对数学其他领域的影响和应用。

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.