Topics in 3-dimensional topology

3 维拓扑主题

• 批准号: 0805942
• 负责人: Efstratia Kalfagianni
• 金额: $ 13.93万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805942&HistoricalAwards=false
• 关键词: Topics; dimensional; topology

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.

该项目涉及从经典的三维拓扑和双曲几何的技术应用到多项式链接不变量和有限型不变量的链接和3-流形的研究：（a）PI已经证明的结果表明双曲链接的体积和琼斯多项式的系数之间的深层联系。 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)调查链接多项式不变量的图论多项式的连接（例如Bollob\'as？Riordan多项式）。她希望在这个框架内研究链接不变量，可以导致链接多项式和Khovanov同调和链接补的几何结构之间的新联系。(c)使用开发的通用机器来理解3-流形在手术下如何变化，以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3-流形理论，组合技术从Dehn手术和表面映射组技术。 该项目的研究在于三维拓扑领域的中心对象的研究是空间称为3流形。 三维流形是一个局部看起来像普通三维空间的对象，但其整体结构可能是复杂的。 三维拓扑学的一个重要部分也是研究结（以某种纠缠的方式嵌入三维流形中的环）及其分类。 拓扑学家解决这些问题的方法之一是使用不变量。近年来，源于物理学的思想引导数学家们发现了纽结和三维流形的各种不变量。PI项目的中心主题是理解这些不变量的性质，使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。