Noncommutative and Heegaard Floer Methods in Low-Dimensional Topology

PI将使用拓扑和代数技术以及Heegaard flower同调和von Neumann代数技术的组合来解决有关三维空间中的结、链路和空间图的特定问题。项目第一部分的目的是通过研究拓扑切片结子群的一种称为双极过滤的新过滤，定义n可解商的初等分解，并将C视为度量空间并研究作用于其上的算子，从而更好地理解光滑结和谐群C。关于结调和群的问题与光滑4流形的问题密切相关，因此将为4流形的神秘和激动人心的世界提供洞察力。在第二个项目中，PI提出建立一个分类高阶和扭曲亚历山大多项式的高阶Heegaard flower同调理论。PI将使用这个理论来定义新的协调不变量，推广Heegaard flower不变量。在第三个项目中，PI将通过PI和合作者最近定义的空间图的图花同调不变量来研究Legendrian空间图和空间图的一致性。结理论是对三维空间中打结的圆或弦的研究。自从开尔文勋爵（错误地）假设原子是由结在一起的乙醚管构成以来，结已经被研究了一个多世纪。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联系的。这个项目将更好地理解结以及它们如何在3维和4维中相互作用。由于我们生活的世界是三维的（如果考虑到时间，则是四维的），结点和链接在许多现实生活中的应用中起着特殊的作用。例如，结的琼斯多项式与统计力学中的波特模型有关。此外，结理论在DNA研究和癌症研究中发挥着重要作用。人们可以把环状DNA看作是一个结，某些酶，称为拓扑异构酶，操纵DNA；这种操作可以看作是结上的某些简单动作。

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.