Geometric Aspects Knot and 3-manifold Invariants

这个由美国国家科学基金会资助的项目的研究是在三维拓扑领域，其中研究的中心对象是被称为三流形的空间。三流形是一个局部看起来像普通三维空间，但其整体结构可能是复杂的空间。三维拓扑学的一个重要部分也是对结的研究，或者换句话说，在三流形中以某种纠缠的方式嵌入的环。Thurston's Geometrization Conjecture（瑟斯顿几何猜想）这个著名问题的解已经证明，三流形和其中的补结点可以分解成若干块，这些块承认显式几何。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。在实际应用中，三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既自然又重要。拓扑学家研究三流形的方法之一是通过构造和研究被称为不变量的对象。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结和三流形的各种微妙而强大的不变量。理解拓扑和组合量以及不变量与详细几何结构之间的联系，是低维拓扑学的一个中心和重要目标。该项目的主要主题是建立这样的联系，并探索其分支和应用到拓扑学以及其他数学领域。该项目旨在建立链路和三流形的几何和拓扑描述、性质和量子不变量之间的内在联系。该项目的一部分将继续PI对琼斯型连杆多项式，连杆补中基本曲面的拓扑结构和双曲几何之间关系的研究。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与双曲几何的联系。第三部分将开发从纯组合输入中识别三流形几何结构的方法，并从拓扑数据中导出几何量的估计。第四部分将研究三流形中的绞链连杆理论，它的不变量，以及它与三流形几何分解的相互作用。该项目还涉及目前在PI工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.