Extensions of Heegaard Floer Homology and Applications to Topology

低维拓扑研究一维到四维空间的形状，其应用范围从物理学和宇宙学（宇宙的形状）到生物化学（了解打结 DNA 的行为）。与 3 维和 4 维空间的研究密切相关的是对结（空间中系结的环）的研究。 2000 年代初期，Ozsvath 和 Szabo 开发了一系列强大的纽结以及 3 维和 4 维空间不变量，通常称为 Heegaard-Floer 同调。 Heegaard-Floer同调从此在低维拓扑学中占据了重要地位，并帮助获得了许多新结果并解决了许多旧猜想。 PI 和她的合作者开发了一种新的代数技术来研究结的变体（结弗洛尔同源性），方法是将结切割成称为缠结的碎片，并研究各个碎片及其粘合。这个 NSF 资助的项目旨在进一步开发这种称为 tangle Floer 同源性的新工具，并用它来研究结理论问题并更好地理解各种结不变量之间的关系。 PI 的目标是让本科生和研究生参与该项目具有组合和计算性质的方面，并将继续向更广泛的受众推广数学。 该项目的具体组成部分是：1）全面开发tangle Floer同调的“负”版本，引入新的一致性不变量，并将tangle Floer同调扩展到整数系数；2）理解和发展knot Floer同调与量子代数之间的联系，以便关联现有的knot不变量并获得新的不变量；3）使用tangle Floer同调在knot Floer中进行计算同源性并研究有关结和链接一致性、变异、弦链接和周期结的问题。

Low-dimensional topology studies the shapes of spaces in dimensions one through four, and has applications ranging from physics and cosmology (the shape of the universe) to biochemistry (understanding the behavior of knotted DNA). Closely related to the study of 3- and 4-dimensional spaces is the study of knots (loops tied in space). In the early 2000s, Ozsvath and Szabo developed a package of powerful invariants for knots and 3- and 4-dimensional spaces, generally known as Heegaard-Floer homology. Heegaard-Floer homology has since taken a major place in low-dimensional topology, and has helped obtain a lot of new results and settle numerous old conjectures. The PI and her collaborator have developed a new algebraic technique for studying the variant for knots (knot Floer homology), by cutting a knot into pieces called tangles, and studying the individual pieces and their gluing. This NSF funded project seeks to develop further this new tool called tangle Floer homology, and use it to study knot theory problems and to better understand the relation between various knot invariants. The PI will aim to involve undergraduate and graduate students in aspects of the project that have combinatorial and computational nature, and will continue to promote mathematics to a broader audience. Specific components of the project are to: 1) fully develop the ''minus" version of tangle Floer homology and introduce new concordance invariants, as well as extend tangle Floer homology to integer coefficients; 2) understand and develop the connections between knot Floer homology and quantum algebra, in order to relate existing knot invariants as well as obtain new ones; 3) use tangle Floer homology to carry out computations in knot Floer homology and to study problems on knot and link concordance, mutation, string links, and periodic knots.