Geometric Aspects Knot and 3-manifold Invariants

这个NSF资助的项目的研究领域是三维拓扑，其中研究的中心对象是称为三维流形的空间。 一个三维流形是一个空间，局部看起来像普通的三维空间，但其整体结构可能很复杂。三维拓扑学的一个重要部分也是研究结，或者换句话说，以某种纠缠方式嵌入三维流形中的环。瑟斯顿几何化猜想（Thurston's Geometrization Conjecture）是一个著名的问题，它的解决方案已经确定了三流形和其中的结点的补数可以分解成允许明确几何的部分。在这种情况下出现的最常见和最有趣的几何之一是双曲几何。 在实践中，三维流形往往是以组合拓扑描述的形式给出的，寻求从这些描述中推导几何信息的方法是自然而重要的。拓扑学家研究三流形的方法之一是通过构造和研究称为不变量的对象。在过去的几十年里，起源于量子物理学的思想使数学家们发现了纽结和三流形的各种微妙而强大的不变量。 了解拓扑和组合的数量和不变量的详细几何结构的连接，从瑟斯顿的图片，是一个中心和重要的目标低维拓扑。 本项目的主题是建立这种联系，并探索其分支和应用拓扑学以及其他数学领域。该项目旨在建立几何和拓扑描述之间的内在联系，属性，以及链接和三维流形的量子不变量。该项目的一部分将继续研究琼斯型链接多项式之间的关系，链接补充和双曲几何中的基本曲面拓扑。另一部分，将研究Turaev-Viro三流形不变量，它们与其他量子不变量的关系以及它们的渐近性与双曲几何的联系。第三部分将开发用于从纯组合输入识别三流形上的几何结构的方法，并从拓扑数据中获得几何量的估计。 第四部分将研究三流形中的绞链理论，它的不变量，以及它与三维流形的几何分解的相互作用。 该项目还涉及目前与PI合作的研究生的研究问题。

The research in this NSF funded project lies in the area of three-dimensional topology, where the central objects of study are spaces called three-manifolds.  A three-manifold is a space that locally looks like the ordinary three-dimensional space but whose global structure may be complicated. An important part of three-dimensional topology is also the study of knots, or in other words, loops embedded in some tangled way in three-manifolds. The solution of a well-known problem known as Thurston's Geometrization Conjecture has established that three-manifolds, and complements of knots in them, decompose into pieces that admit explicit geometries. One of the most common and most interesting geometries that appear in this setting is hyperbolic geometry.  In practice, three-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 three-manifolds is through the construction and study of objects called invariants. In the last few decades, ideas that originated in quantum physics have led mathematicians to the discovery of a variety of subtle and powerful invariants of knots and three-manifolds.  Understanding the connections of topological and combinatorial quantities and invariants to detailed geometric structures, arising from Thurston's picture, is a central and important goal of low dimensional topology.  The main theme of this project is to establish such connections and to explore their ramifications and applications to topology  as well as other areas of mathematics.The project aims to establish intrinsic connections between geometry and topological descriptions, properties, and quantum invariants of links and three-manifolds. One part of the project will continue the PI's study of the relations between Jones-type link polynomials, the topology of essential surfaces in link complements and hyperbolic geometry. Another part, will study the Turaev-Viro three-manifold invariants, their relations to other quantum invariants and the connections of their asymptotics to hyperbolic geometry. A third part will develop methods for recognizing geometric structures on three-manifolds from purely combinatorial input, and derive estimates on geometric quantities from topological data.  A fourth part will study skein link theory in three-manifolds, its invariants, and its interaction with geometric decompositions of 3-manifolds.  The project also involves research problems  for graduate students currently working with the PI.