Knot and 3-Manifold Invariants, Seifert Surfaces and Dehn Surgery

结和 3 流形不变量、Seifert 曲面和 Dehn 手术

• 批准号: 0104000
• 负责人: Efstratia Kalfagianni
• 金额: $ 5.8万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-08-15 至 2004-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104000&HistoricalAwards=false
• 关键词: Knot; Manifold; Invariants; Seifert; Surfaces

AbstractAward: DMS-0104000Principal Investigator: Efstratia KalfagianniThe PI will study the ``finite type" invariants of knots and3-manifolds by using techniques from classical 3-dimensionaltopology and search for geometric information encoded in theseinvariants. First, she will continue her work on developing thetheory of finite type invariants for knots and links in arbitrary3-manifolds by using techniques from the theory of atoroidaldecompositions of 3-manifolds. She also plans to work onrelating the Vassiliev knot invariants to properties of Seifertsurfaces spanned by the knots and to intrinsic invariants of theknot complement. This will extend classical results about thetopology of the Alexander polynomial to the Jones polynomial andits generalizations. Finally, the PI plans to search forrelations between the Jones polynomial of a knot and thefundamental group of 3-manifolds obtained by Dehn surgery on theknot.The research of the project lies in the area of 3-dimensionaltopology the central objects of study of which are spaces called3-manifolds. A 3-manifold is an object that locally looks likethe ordinary 3-dimensional space but whose global structure canbe complicated. A main goal of 3-dimensional topology is tounderstand these structures and achieve a classification of3-manifolds. An important part of 3-dimensional topology is alsothe study of knots (loops embedded in some tangled way in3-manifolds) and their classification. One of the ways thattopologists have been approaching these problems is through theuse of ``invariants". In the recent years, ideas originated inphysics, lead mathematicians to the discovery of a variety ofinvariants of knots and 3-manifolds. The central theme of thePI's project is to understand the properties of these invariants,using ideas from traditional 3-dimensional topology and fromphysics, and to look for applications to the aforementionedclassification problems.

Abstractaward：DMS-0104000原理研究者：Efstratia kalfagiannithe Pi将研究``有限类型''''nong和3 manifolds的不变式和3个manifolds，并使用经典的3-二维论的技术使用来自经典的3-二维的技术，并搜索以这些范围编码的几何信息，她将继续进行。通过使用3个manifords的atoroidaldectosigons的技术，她计划在塞氏菌的属性上进行链接。最终，PI计划在结节的琼斯多项式和Dehn手术对该项目的研究中搜索的福音。一个3个manifold是一个当地看起来像普通的3维空间但其全球结构可能复杂的对象。 三维拓扑结构的主要目标是Toundersters制定这些结构，并实现3个manifolds的分类。三维拓扑结构的重要组成部分是对结的研究（以某种纠结的方式嵌入了3个体中的循环）及其分类。浮游物学家一直在处理这些问题的一种方式是通过``不变式''的使用。在近年来，想法起源于非物理学，导致数学家发现结的各种各样的结和3个manifolds。上述级别的问题。