Collaborative research: Hyperbolic geometry of knots and 3-manifolds

佩雷尔曼几何化定理没有回答的一个中心问题是，3流形的组合特征究竟应该如何与它的几何结构联系起来。首席研究员Futer和Purcell将研究这个问题的几个方面。本项目的第一个目标是使用与曲面相关的组合复合体对纤维双曲3流形的几何形状给出显式估计。第二个密切相关的目标是使用一般结或链接的辫子表示，以便对其补充的体积给出明确的图解估计。第三，pi将继续他们与Kalfagianni的联合项目，将结和链接互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，调查人员将继续与库珀合作，了解解开隧道的几何特性。3流形是一个空间，在这个空间里，像直升机这样的物体可以在三个不同的垂直方向上移动。我们所居住的宇宙是一个三流形，其整体几何我们还不了解。另一个丰富的例子来自不同结周围的空间。Thurston， Perelman和Mostow的强大定理表明，几乎每个结补，更一般地说，几乎每个3流形，都有一个唯一的双曲度规。也就是说，有一种标准的方法来测量空间，使每一个二维截面曲线都像一个马鞍。目前，虽然我们知道这个标准双曲度规存在，但我们对如何将它与容易计算的量（如结图的复杂性）联系起来知之甚少。这个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，pi所研究的几何问题是非常直观和动手的，自然地衍生为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.