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：使用几何。不久，人们注意到双曲流形形成了最大也是最不为人所知的流形类别。非正式地说，双曲三维流形是在具有双曲度规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-球面中的纽结和链补。此外，纽结和链环本身就是一个研究对象，纽结理论在纯数学和应用研究领域都有许多应用。节点很容易绘制为平面图，其中包含有关交叉路口的地下通道/立交桥的信息。然而，建立图和相应的三维流形的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和来自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帮助使用几何透视的软件，并将继续这些活动。与M.Thilethwaite一起，PI提出了一种直接从链接图计算双曲线结构的新方法，朝着第一个目标取得了进展。与W.Neumann一起，她将这种方法推广到参数化尖三维流形的双曲结构。这个项目将新方法与各种技术相结合，系统地研究了双曲连杆的内在几何及其与组合图像的联系。在所考虑的公开问题中，有关于正则单元分解、双曲线体积、各种圆弧的长度、几何三角剖分等的问题。虽然这些问题是先验的有趣的，但该项目的一个重要部分与所获得的几何洞察的即时应用有关。例如，PI将探索几何不变量和量子链接不变量之间的关系，研究双曲三维流形的几何不变量和算术不变量之间的联系，使用几何和计算机计算来处理纠缠表，以及解决其他一些未决问题。

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.