Hyperbolic Structures from Link Diagrams

瑟斯顿的几何化猜想促使人们从一个新的角度来研究拓扑学中的一个主要对象--流形。很快人们注意到双曲流形形成了最大和最不了解类流形。非正式地，双曲3-流形是在具有双曲度量的三维空间上局部建模的对象。这类流形的一个重要子类是3-球面中的纽结补和环补。此外，纽结和链环本身也是一个研究对象，纽结理论在纯数学和应用研究领域都有许多应用。一个结很容易画成一个平面图，上面有关于交叉口处的地下通道/立交桥的信息。然而，建立相应的三维流形的图形和几何性质之间的联系是一项艰巨的任务。这是该提案的第一个主要目标。另一个目标是探索几何不变量与其他数学领域的不变量之间的关系。PI致力于指导学生研究和开发有助于使用几何透视的软件，并将继续这些活动。Thistlethwaite，PI通过提出一种直接从链接图计算双曲结构的新方法，朝着第一个目标取得了进展。与W.诺依曼，她推广了这种方法参数双曲结构的尖3流形。该项目将新方法与各种技术相结合，用于系统研究双曲连杆的内在几何及其与组合图片的连接。在开放的问题中考虑的问题是典型的细胞分解，双曲量，各种弧的长度，几何三角形等，而这些问题是有趣的先验，该项目的一个重要部分是关注的几何洞察力获得的直接应用。例如，PI将探索几何和量子连接不变量之间的关系，研究双曲3流形的几何和算术不变量之间的联系，使用几何和计算机计算来处理缠结列表，并解决其他一些悬而未决的问题。

Thurston's Geometrization conjecture prompted the study of one of the main objects in topology, a manifold, from a new perspective: using geometry. Soon it was noticed that hyperbolic manifolds formed the largest and the least understood class of manifolds. Informally, a hyperbolic 3-manifold is an object modeled locally on 3-dimensional space that has hyperbolic metric. An important subclass of such manifolds are knot and link complements in 3-sphere. Additionally, knots and links are an object of study on their own, and knot theory has a number of applications in pure mathematics, as well as in applied fields of study. A knot is easy to draw as a planar diagram with information about underpasses/overpasses at crossings. However, establishing the connection between the diagram and geometric properties of the corresponding 3-manifold is a hard task. This is the first main goal of the proposal. The other goal is exploring the relation between geometric invariants and invariants that come from other areas of mathematics. The PI is engaged in mentoring student research and developing software that helps to use the geometric perspective, and will continue these activities.With M. Thistlethwaite, the PI made progress towards the first goal by suggesting a new method for computing the hyperbolic structure directly from a link diagram. With W. Neumann, she generalized this approach to parameterize hyperbolic structure of a cusped 3-manifold. This project blends the new method with various techniques for a systematic study of the intrinsic geometry of hyperbolic links and its connection with a combinatorial picture. Among the open problems considered are questions about the canonical cell decomposition, hyperbolic volume, lengths of various arcs, geometric triangulations, etc. While these questions are interesting a priori, a significant part of the project is concerned with immediate applications of the geometric insight obtained. For example, the PI will explore the relation between geometric and quantum link invariants, investigate the connection between the geometry and arithmetic invariants of hyperbolic 3-manifolds, approach tangle tabulation using geometry and computer calculations, and tackle some other open questions.