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

摘要奖：DMS-0104000首席研究员：Efstra Kalfagianni PI将使用经典三维拓扑学中的技术研究纽结和3-流形的“有限类型”不变量，并寻找这些不变量中编码的几何信息。首先，她将继续她的工作，通过使用三维流形的拟态分解理论的技巧，发展任意三维流形中纽结和链环的有限类型不变量理论。她还计划将Vassiliev纽结不变量与纽结跨越的Seifert曲面的性质以及纽结补的内在不变量联系起来。这将把有关Alexander多项式拓扑学的经典结果推广到Jones多项式及其推广。最后，PI计划寻找纽结的Jones多项式与通过Dehn手术得到的三维流形的基本群之间的关系。本课题的研究涉及三维拓扑领域，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但其全局结构可能复杂的对象。三维拓扑学的一个主要目标是理解这些结构，并实现对三维流形的分类。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。物理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纽结和三维流形的各种变体。PI项目的中心主题是利用传统的三维拓扑学和物理学的思想来理解这些不变量的性质，并寻找对上述分类问题的应用。

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.