Invariants and geometry of knots and 3-manifolds

该项目解决几何，拓扑和3-流形的组合不变量之间的关系。PI将建立链接和3-流形的几何和拓扑描述，属性和量子不变量之间的具体联系。该项目的一部分将结合联合收割机几种技术，开发 PI和她的合作者以及其他人，从Dehn手术演示，链接图和群作用等拓扑数据中获得双曲体积和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），本质表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，它的不变量，并研究它与来自几何化图像的详细结构的相互作用。本研究也涉及目前与PI合作的研究生。该项目的研究领域在于三维拓扑结构。本研究的中心对象是称为三维流形的空间。三维流形是一个局部看起来像普通三维空间的对象，但其整体结构可能是复杂的。三维拓扑学的一个重要部分也是研究结（以某种纠缠的方式嵌入三维流形中的环）及其分类。瑟斯顿的几何化猜想的解决方案已经建立了3-流形（和其中的结的补）分解成允许显式几何的部分，而双曲几何是更经常出现的几何。然而，在实践中，3-流形往往给出的组合拓扑描述，这是自然的，重要的是寻求方法来推导几何信息，从这些描述。拓扑学家研究三维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各种各样的纽结和三维流形的不变量。理解拓扑和组合量和几何不变量的连接是三维拓扑学的中心和重要目标。这个项目的主题是建立具体的这种联系，并探讨其影响和应用到其他领域的数学。

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.