Heegaard Floer homology, concordance, and categorification

Heegaard Floer 同源性、一致性和分类

• 批准号: 1642577
• 负责人: Jennifer Hom
• 金额: $ 1.27万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1642577&HistoricalAwards=false
• 关键词: Heegaard; Floer; homology; concordance; categorification

The principal investigator will use tools from Heegaard Floer homology to study phenomena in low-dimensional topology. One goal is to better understand the knot concordance group, in particular the difference between the smooth and topological categories. Concordance invariants coming from Floer homology, such as d-invariants and epsilon, are well-suited for this task. Specifically, the principal investigator aims to show that the subgroup of the smooth concordance group generated by topologically slice knots contains an infinite rank summand. A different approach to concordance is via bordered Floer homology. In the same sense that knot Floer homology categorifies the Alexander polynomial, one may hope that the bordered Floer homology of a Seifert surface categorifies the Seifert form. Many classical concordance invariants can be defined in terms of the Seifert form, and the project will investigate how these invariants manifest themselves within the bordered invariants.The structure of knotted curves in space is intimately related to our understanding of the shape of 3- and 4-dimensional space. Thinking of time as the fourth dimension, the study of knot concordance becomes a question about the evolution of knots over time. By putting different restrictions on how knots may evolve, one obtains different definitions of the complexity of a knot. These measures of complexity have applications from the very small (e.g., the behavior of knotted strands of DNA) to the very large (e.g., the shape of the universe).

首席研究员将使用 Heegaard Floer 同调工具来研究低维拓扑中的现象。目标之一是更好地理解结索引组，特别是平滑类别和拓扑类别之间的差异。来自 Floer 同源性的一致性不变量（例如 d 不变量和 epsilon）非常适合此任务。具体来说，主要研究者的目的是证明由拓扑切片结生成的平滑索引群的子群包含无限秩被加数。另一种不同的一致性方法是通过有界弗洛尔同源性。在结 Floer 同调对 Alexander 多项式进行分类的相同意义上，人们可能希望 Seifert 曲面的有界 Floer 同调对 Seifert 形式进行分类。许多经典的一致性不变量可以用 Seifert 形式来定义，该项目将研究这些不变量如何在有边界不变量中表现出来。空间中打结曲线的结构与我们对 3- 和 4 的形状的理解密切相关维空间。将时间视为第四个维度，对绳结一致性的研究就变成了关于绳结随时间演变的问题。通过对结的演化方式施加不同的限制，人们可以获得对结复杂性的不同定义。这些复杂性测量的应用范围从非常小（例如，DNA 打结链的行为）到非常大（例如，宇宙的形状）。