Topics in 3-dimensional topology

本项目涉及经典三维拓扑和双曲几何技术在研究多项式连杆不变量和连杆和3流形的有限型不变量方面的应用：(a) PI证明了双曲连杆的体积与其琼斯多项式的系数之间的深刻联系。她想进一步研究这些联系，并在量子拓扑和双曲几何之间建立一座桥梁。(b)研究链接多项式不变量与图论多项式(例如Bollob\'as ？赖尔登多项式)。她希望在这个框架内研究连杆不变量，可以在连杆多项式与Khovanov同调和连杆补的几何结构之间建立新的联系。(c)利用为理解3流形在手术中如何变化而开发的通用机制来探索有限型结不变量所捕获的拓扑关系。这里的主要工具包括缝合的三流形理论，Dehn手术的组合技术和表面映射群技术。该项目的研究是在三维拓扑领域，其研究的中心对象是被称为3流形的空间。三维流形是一种局部看起来像普通三维空间的物体，但其整体结构可能很复杂。三维拓扑学的一个重要部分也是对结点（以某种纠缠方式嵌入在三维流形中的环）及其分类的研究。拓扑学家解决这些问题的方法之一就是使用不变量。近年来，源于物理学的思想，使数学家们发现了结和3流形的各种不变量。PI项目的中心主题是了解这些不变量的性质，利用三维拓扑和几何以及物理学的思想，并研究它们区分结和3流形的程度。

This project concerns applications of techniques from classical 3-dimensional topology and hyperbolic geometry to the study of polynomial link invariants and finite type invariants of links and 3-manifolds:(a) The PI has proved results suggesting deep connections between the volume of hyperbolic links and the coefficients of their Jones polynomial. She would like to further investigate these connections and establish a bridge between quantum topology and hyperbolic geometry. (b) Investigate the connections of link polynomial invariants to a graph theoretic polynomials (e.g. the Bollob\'as ?Riordan polynomial). She hopes that studying link invariants within this framework, can lead to new connections between link polynomials and Khovanov homology and geometric structures of link complements.(c) Use the general machinery developed to understand how 3-manifolds change under surgery to explore the topological relations captured by the finite type knot invariants. The main tools here include sutured 3-manifold theory, combinatorial techniques from Dehn surgery and surface mapping group techniques. The research of the project lies in the area of 3-dimensional topology the central objects of study of which are spaces called 3-manifolds. A 3-manifold is an object that locally looks like the ordinary 3-dimensional space but whose global structure can be complicated. An important part of 3-dimensional topology is also the study of knots (loops embedded in some tangled way in 3-manifolds) and their classification. One of the ways that topologists have been approaching these problems is through the use of invariants. In the recent years, ideas originated in physics, lead mathematicians to the discovery of a variety of invariants of knots and 3-manifolds. The central theme of the PI's project is to understand the properties of these invariants, using ideas from 3-dimensional topology and geometry and from physics, and investigate the extent to which they distinguish knots and 3-manifolds.