Homological Invariants of Knots and Three-Manifolds

结和三流形的同调不变量

• 批准号: 0603940
• 负责人: Zoltan Szabo
• 金额: $ 12.15万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603940&HistoricalAwards=false
• 关键词: Homological; Invariants; Knots; Three; Manifolds

DMS-0603940Jacob RasmussenThis project is about a class of knot invariants known as ``knot homologies'' which generalize classical invariants such as theAlexander and Jones polynomial. The PI plans to investigate a conjectured relationship between two versions of these invariants. The first version was developed by Khovanov and Rozansky and is combinatorial in nature. The second is known as knot Floer homology, and has its origins in the Heegaard Floer homology of Ozsvath and Szabo. It is not combinatorial, but is known to carry a great deal of geometric information about the knot. Despite the differences in their definitions, these two theories exhibit some truly striking similarities. The project aims to explain these similarities and to use them to get a better understanding of both theories. One posssible application is to find combinatorial analogs of gauge theoretic invariants like the knot Floer homology.The study of knotted curves in three-dimensional space is intimately related to the geometry of three- and four-dimensional spaces themselves. In the last few decades, ideas from physics have led to the development of two major types of invariants for such curves, known as ``quantum'' and ``gauge theoretic'' invariants. Although both types have roots in the quantum theory of fields, there was little sign that they were related. Recently, however, some remarkable similarities between the two have begun to appear. The aim of this project is to better understand the relationship between these two classes of invariants.

DMS-0603940JACOB RASMUSSENTHIS项目是关于一类称为``结''''''''的结，概括了诸如Thealexander和Jones多项式的经典不变性。 PI计划研究这些不变的两个版本之间的猜想关系。第一个版本是由Khovanov和Rozansky开发的，本质上是合并的。第二个被称为结式同源性，起源于Ozsvath和Szabo的Heegaard Floer同源性。它不是组合，而是众所周知，它具有有关结的大量几何信息。尽管其定义有所不同，但这两种理论表现出一些真正的相似之处。该项目旨在解释这些相似之处，并使用它们来更好地了解这两种理论。一种可能的应用是找到仪表理论不变的组合类似物，例如结式浮点同源性。三维空间中的打结曲线的研究与三维空间本身的几何形状密切相关。在过去的几十年中，物理学的想法导致了这种曲线的两种主要类型的不变性，称为``量子'''和````Quantum''''和``衡量理论''。尽管两种类型都在量子的田间理论中根源，但几乎没有迹象表明它们是相关的。然而，最近，两者之间的一些显着相似之处已经开始出现。该项目的目的是更好地了解这两个类别不变的人之间的关系。