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结构和双曲线几何形状到多项式链接不变性和有限型链接和3个manifolds的有限类型不变性的技术的应用： 她想进一步研究这些连接，并在量子拓扑和双曲线几何形状之间建立桥梁。 （b）研究链接多项式不变的连接与图理论多项式（例如，bollob \'as？riordan多项式）。她希望在此框架内研究链接不变性，可以导致链接多项式与Khovanov同源性和链接补充的几何结构之间的新联系。（c）使用开发的通用机械来了解手术下的3个模型如何在手术下变化，以探索有限的拓扑关系由有限的拓扑类型的小玩意生群体捕获。这里的主要工具包括缝合的3个manifold理论，来自Dehn手术和表面映射组技术的组合技术。 该项目的研究在于三维拓扑区域的研究中心对象，其空间称为3个manifolds。 一个3个manifold是一个本地看起来像普通的3维空间，但其整体结构可能复杂的对象。 三维拓扑结构的重要组成部分是对结的研究（嵌入了3个manifolds中的某种纠结的循环）及其分类。 拓扑师一直在处理这些问题的方式之一是使用不变性。近年来，想法起源于物理学，导致数学家发现各种不变的结和3个manifolds。 PI项目的中心主题是使用三维拓扑和几何形状以及物理学的想法来了解这些不变的属性，并研究它们区分结和3个manifolds的程度。