Collaborative research: Hyperbolic geometry of knots and 3-manifolds

佩雷尔曼几何化定理的一个中心问题是，一个3-流形的组合特征与它的几何特征之间到底有什么关系。主要研究者Futer和珀塞尔将研究这个问题的几个方面。本项目的第一个目标是利用曲面的组合复形给出纤维双曲三维流形的几何性质的显式估计。第二，密切相关的，目标是使用辫子介绍一个通用的结或链接，以提供明确的，图解估计其补充量。第三，PI将继续他们与Kalfagianni的联合项目，将结和链补的几何拓扑与量子不变量（如有色琼斯多项式）联系起来。最后，研究人员将继续与库珀的联合工作，以了解解开隧道的几何特性。三维流形是一个空间，其中一个物体，如直升机，可以在三个不同的垂直方向上移动。我们居住的宇宙是一个三维流形，我们还不了解它的整体几何。另一个丰富的例子来自不同节点周围的空间。Thurston、Perelman和Mostow的强有力的定理意味着几乎每一个纽结补，更一般地说，几乎每一个3-流形，都有一个唯一的双曲度量。也就是说，有一个标准的方法来测量空间，使每个二维横截面像马鞍一样弯曲。目前，虽然我们知道这个标准的双曲度量存在，但对于如何将它与易于计算的量（如纽结图的复杂性）联系起来，我们知之甚少。本项目的主要目标是使这些关系更加具体。这个项目的一个重要特点是，由PI研究的几何问题是非常直观和动手，与自然的副产品到软件应用程序和学生的项目。

One central question left unanswered by Perelman's geometrization theorem is exactly how the combinatorial features of a 3-manifold should relate to its geometry. The principal investigators Futer and Purcell will study several aspects of this question. The first goal of this project is to use the combinatorial complexes associated a surface to give explicit estimates on the geometry of a fibered hyperbolic 3-manifold. A second, closely related, goal is to use braid presentations of a generic knot or link in order to give explicit, diagrammatic estimates on the volume of its complement. Third, the PIs will continue their joint project with Kalfagianni to relate the geometric topology of knot and link complements to quantum invariants such as the colored Jones polynomials. Finally, the investigators will continue their joint work with Cooper to understand the geometric properties of unknotting tunnels. A 3-manifold is a space where an object such as a helicopter can move around in three distinct perpendicular directions. The universe that we inhabit is a 3-manifold whose global geometry we do not yet understand. Another rich source of examples comes from the spaces that surround different knots. Powerful theorems of Thurston, Perelman, and Mostow imply that almost every knot complement, and more generally almost every 3-manifold, has a unique hyperbolic metric. That is, there is a standard way to measure the space, so that every 2-dimensional cross-section curves like a saddle. At the moment, while we know that this standard hyperbolic metric exists, very little is known about how to relate it to easily computable quantities such as the complexity of a knot diagram. The main goal of this project is to make these relations much more concrete. One important feature of this project is that the geometric problems studied by the PIs are very visual and hands-on, with natural spin-offs into both software applications and projects for students.