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

项目负责人：Efstratia kalfagiannipi将利用经典三维拓扑技术研究结点和3流形的“有限型”不变量，并搜索这些不变量中编码的几何信息。首先，她将继续她的工作，通过利用3-流形的反流分解理论的技术，发展任意3-流形中结点和连杆的有限型不变量理论。她还计划将瓦西里耶夫结不变量与结张成的塞弗曲面的性质以及结补的固有不变量联系起来。这将把关于Alexander多项式拓扑的经典结果推广到Jones多项式及其推广。最后，PI计划寻找结的琼斯多项式与结上的Dehn手术得到的3流形基本群之间的关系。该项目的研究是在三维拓扑领域，其研究的中心对象是被称为三维流形的空间。三维流形是一个局部看起来像普通三维空间的物体，但其整体结构可能很复杂。三维拓扑学的一个主要目标是理解这些结构并实现三维流形的分类。三维拓扑学的一个重要部分也是对结点（在三维流形中以某种纠缠方式嵌入的环）及其分类的研究。拓扑学家解决这些问题的方法之一是使用“不变量”。近年来，源于物理学的思想使数学家们发现了结和3流形的各种变体。pi项目的中心主题是了解这些不变量的性质，使用传统的三维拓扑和物理学的思想，并寻找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.