Noncommutative and Heegaard Floer Methods in Low-Dimensional Topology

低维拓扑中的非交换和 Heegaard Florer 方法

• 批准号: 1309070
• 负责人: Shelly Harvey
• 金额: $ 24.88万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-05-15 至 2017-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309070&HistoricalAwards=false
• 关键词: Noncommutative; Heegaard; Floer; Methods; Low

The PI will resolve specific questions about knots, links, and spatial graphs in 3-dimensional space using a combination of topological and algebraic techniques as well as techniques from Heegaard Floer homology and von Neumann algebras. The aim of the first part of the project is to gain a better understanding of the smooth knot concordance group C, by studying a new filtration of the subgroup of topologically slice knots called the bipolar filtration, defining a primary decomposition for the n-solvable quotients, and viewing C as a metric space and studying operators acting on it. Questions about the knot concordance group are closely related to questions about smooth 4-manifolds and thus will provide insight into the mysterious and exciting world of 4-manifolds. In the second project, the PI proposes to establish a higher-order Heegaard Floer homology theory that categorifies the higher-order and twisted Alexander polynomials. The PI will use this theory to define new concordance invariants generalizing the Heegaard Floer tau invariant. In the third project, the PI will investigate the study Legendrian spatial graphs and spatial graph concordance via the recent Graph Floer homology invariant of spatial graph defined by the PI and coauthor. Knot theory is the study of knotted circles or strings in 3-dimensional space. Knots have been studied for over a century, ever since Lord Kelvin (incorrectly) hypothesized that atoms were made of knotted tubes of ether. However, we are still far from a complete classification of them or how they are related to one another. This project will give a better understanding of knots and how they interact in 3- and 4-dimensions. Since the world we live in is 3-dimensional (or 4-dimensional if one considers time), knots and links play a special role in many real life applications. For example, the Jones polynomial of a knot is related to the Pott's model in statistical mechanics. In addition, knot theory plays an role in the study of DNA and cancer research. One can view circular DNA as a knot and certain enzymes, called topoisomerases, manipulate DNA; this manipulation can be viewed as certain simple moves on knots.

PI将使用拓扑和代数技术以及Heegaard Floer同调和von Neumann代数的技术相结合来解决三维空间中有关结，链接和空间图的具体问题。 该项目第一部分的目的是更好地理解光滑纽结和谐群C，通过研究拓扑切片纽结子群的一种新的过滤，称为双极过滤，定义了n-可解同胚的初等分解，把C看作度量空间，研究作用在C上的算子，纽结协调群的问题与光滑性问题密切相关，4-流形，从而将提供洞察神秘和令人兴奋的世界4-流形。 在第二个项目中，PI建议建立高阶Heegaard Floer同调理论，将高阶和扭曲的亚历山大多项式归类。 PI将使用该理论来定义新的一致性不变量，从而推广Heegaard Floer tau不变量。在第三个项目中，PI将通过PI和合著者最近定义的空间图的Graph Floer同调不变量来研究Legendrian空间图和空间图一致性。纽结理论是研究三维空间中的打结的圆或弦。 自从开尔文勋爵（错误地）假设原子是由以太的打结管组成以来，人们对结的研究已经超过了一个世纪。 然而，我们还远远没有对它们进行完整的分类，或者它们之间是如何相互关联的。 这个项目将提供一个更好的了解结，以及他们如何在3-和4-维相互作用。 由于我们生活的世界是三维的（如果考虑时间的话是四维的），结和链接在许多真实的生活应用中扮演着特殊的角色。 例如，纽结的琼斯多项式与统计力学中的波特模型有关。 此外，纽结理论在DNA研究和癌症研究中也发挥着重要作用。 人们可以把环状DNA看作是一个结，而某些酶，称为拓扑异构酶，可以操纵DNA;这种操纵可以被看作是在结上的某些简单移动。