Collaborative research: Hyperbolic geometry of knots and 3-manifolds

合作研究：结和三流形的双曲几何

• 批准号: 1007437
• 负责人: Jessica Purcell
• 金额: $ 14.4万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007437&HistoricalAwards=false
• 关键词: Collaborative; research; Hyperbolic; geometry; knots

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.

佩雷尔曼几何化定理未回答的一个核心问题是，3 流形的组合特征应如何与其几何相关。主要研究者 Futer 和 Purcell 将研究这个问题的几个方面。该项目的第一个目标是使用与表面相关的组合复合体来对纤维双曲 3 流形的几何形状进行明确的估计。第二个密切相关的目标是使用通用结或链接的编织表示，以便对其补体的体积给出明确的、图形化的估计。第三，PI 将继续与 Kalfagianni 的联合项目，将结和链接补体的几何拓扑与量子不变量（例如彩色琼斯多项式）联系起来。最后，研究人员将继续与库珀合作，了解解开隧道的几何特性。 三歧管是一个空间，其中直升机等物体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流形，我们还不了解其整体几何形状。另一个丰富的例子来源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，都具有唯一的双曲度量。也就是说，有一个标准的方法来测量空间，使得每个二维横截面都像马鞍一样弯曲。目前，虽然我们知道这种标准双曲度量的存在，但对于如何将其与易于计算的量（例如结图的复杂性）联系起来却知之甚少。该项目的主要目标是使这些关系更加具体。该项目的一个重要特点是，PI 研究的几何问题非常直观和实用，可以自然地衍生到软件应用程序和学生项目中。