Topics in 3-dimensional topology

这个项目涉及经典的三维拓扑学和双曲几何在研究多项式链环不变量和链环和三维流形的有限类型不变量方面的应用：(A)PI证明了双曲链环的体积和它们的Jones多项式的系数之间存在着深刻的联系。她想进一步研究这些联系，并在量子拓扑学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，作为Riordan多项式的Bollob)之间的联系。她希望在这个框架内研究链环不变量，可以在链环多项式与Khovanov同调和链环补的几何结构之间产生新的联系。(C)利用开发的理解三维流形在手术下如何变化的通用机制来探索有限类型结点不变量捕捉到的拓扑关系。这里的主要工具包括缝合的三维流形理论、Dehn手术的组合技术和表面标测组技术。本课题的研究领域是三维拓扑学，其中心研究对象是称为三维流形的空间。三维流形是一种局部看起来像普通三维空间的对象，但其全局结构可能很复杂。三维拓扑学的一个重要部分也是对纽结(以某种纠缠方式嵌入到三维流形中的环)及其分类的研究。拓扑学家处理这些问题的方法之一就是使用不变量。近年来，思想起源于物理学，导致数学家们发现了各种纽结和三维流形的不变量。PI项目的中心主题是利用来自三维拓扑和几何以及物理的思想来理解这些不变量的性质，并调查它们区分纽结和三维流形的程度。

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.