Extensions of Heegaard Floer Homology and Applications to Topology

低维拓扑学研究1到4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（了解结DNA的行为）。与对三维和四维空间的研究密切相关的是对结（在空间中捆绑的环）的研究。在21世纪初，Ozsvath和Szabo为结点和3维和4维空间开发了一套强大的不变量，通常被称为Heegaard-Floer同调。heegaard - flower同调从此在低维拓扑中占据了重要地位，并帮助获得了许多新的结果和解决了许多旧的猜想。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（结花同源性），通过将一个结切成称为缠结的片段，并研究单个片段及其粘合。这个由美国国家科学基金会资助的项目旨在进一步发展这个名为“缠结花同源性”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项目中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。项目的具体组成部分是：1)充分发展了缠结Floer同调的“负”版本，引入新的一致性不变量，将缠结Floer同调扩展到整系数；2)理解和发展结花同调与量子代数之间的联系，以便关联现有的结不变量并获得新的结不变量；3)利用缠结花同源性对结花同源性进行计算，研究结链一致性、突变、串链、周期结等问题。

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.