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Extensions of Heegaard Floer Homology and Applications to Topology

Heegaard Floer 同调的扩展及其在拓扑中的应用

基本信息

批准号：
1711100
负责人：
Ina Petkova
金额：
$ 21.03万
依托单位：
Dartmouth College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1711100&HistoricalAwards=false
关键词：
Extensions Heegaard Floer Homology Applications

项目摘要

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.

低维拓扑学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（理解打结的DNA的行为）。与三维和四维空间的研究密切相关的是对结（空间中的环）的研究。在21世纪初，Ozsvath和Szabo开发了一套用于纽结和3维和4维空间的强大不变量，通常称为Heegaard-Floer同调。Heegaard-Floer同调在低维拓扑学中占有重要的地位，并帮助获得了许多新的结果，解决了许多旧的问题。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Floer同源性），通过将结切割成称为缠结的碎片，并研究单个碎片及其粘合。这个NSF资助的项目旨在进一步开发这种称为tangle Floer homology的新工具，并使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具有组合和计算性质的项目，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）充分发展tangle Floer同调的“减”版本，并引入新的协调不变量，以及将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，以便将现有的knot不变量联系起来，并获得新的knot不变量; 3）利用tangle Floer同调进行纽结Floer同调的计算，研究纽结与链环的协调、突变、串链环、周期纽结等问题。