CAREER: Hyperbolic geometry and knots and links

自 20 世纪 80 年代初以来，人们就知道结和链接互补体分解成具有几何结构的碎片，最常见的几何形状是双曲线。 然而，双曲几何与其他结和链接不变量之间的联系仍然没有得到很好的理解。研究人员将利用三流形几何和拓扑的最新发展和技术来建立双曲几何和链接不变量之间的联系，重点关注两个领域的问题。 首先，她将把双曲几何与量子不变量联系起来，特别是继续她最近的工作，寻找双曲几何和有色琼斯多项式之间的联系。 其次，她将根据结和链接的图解和拓扑不变量获得双曲量的界限，例如边界体积和尖点体积，并查找测地线的同位素类。 作为这项工作教育部分的一部分，她将组织两次关于双曲几何联系的会议，并继续与本科生、研究生和 K-12 学生一起工作。该项目涉及称为 3 流形的 3 维空间的研究，其中包括我们的宇宙空间（具有三个空间维度）。 这些空间出现在物理学、力学、微生物学和化学中，因此我们希望更好地理解它们的数学特性。 研究 3 流形的一种方法是从 3 维球体中围绕圆圈钻出管子，然后以不同的方式重新连接管子。 钻孔管绕圆的空间称为链补，如果只有一个圆，则称为结补。 这样，结和连杆互补体就成为 3 流形的构建块。 研究人员将研究结和连接互补体的几何形状，希望找到关于更广泛的 3 流形类别的属性的新结果。 该项目包括许多培训学生的规定。 大部分研究将在本科生和研究生的协助下进行。 此外，作为该项目的一部分，研究者将举办两次研究会议，期间将邀请几位研究生和博士后研究人员介绍他们的相关工作。 最后，研究人员将在整个学年继续为儿童举办数学研讨会。

It has been known since the early 1980s that knot and link complements decompose into pieces admitting a geometric structure, with the most common geometry being hyperbolic. However, the connection between hyperbolic geometry and other knot and link invariants is still not well understood. The investigator will use recent developments and techniques in 3-manifold geometry and topology to make connections between hyperbolic geometry and link invariants, focusing on problems in two areas. First, she will relate hyperbolic geometry to quantum invariants, in particular continuing her recent work to find connections between hyperbolic geometry and the colored Jones polynomial. Second, she will obtain bounds on hyperbolic quantities based on diagrammatical and topological invariants of knots and links, for example bounding volume and cusp volume, and finding isotopy classes of geodesics. As part of the educational portion of this work, she will organize two conferences on connections of hyperbolic geometry, and continue her work with undergraduate, graduate, and K-12 students.This project concerns the study of 3-dimensional spaces called 3-manifolds, which include the space of our universe (with three spatial dimensions). These spaces appear in physics, mechanics, microbiology, and chemistry, and so we wish to better understand their mathematical properties. One way to study 3-manifolds is to drill out tubes around circles from a 3-dimensional sphere, and then reattach the tubes in a different manner. The space of drilled tubes about circles is called a link complement, or knot complement if there is just one circle. In this way, knot and link complements are building blocks for 3-manifolds. The investigator will study the geometry of knot and link complements, with the hope of finding new results on the properties of broader classes of 3-manifolds. This project includes many provisions for training students. Much of the research will be carried out with the assistance of undergraduate and graduate students. In addition, as part of this project the investigator will run two research conferences, during which several graduate students and postdoctoral researchers will be invited to present their related work. Finally, the investigator will continue to run mathematical workshops for children throughout the school year.