Contact Topology, Knots, and Heegaard Floer Theory

摘要奖：DMS-0805836首席研究员：Olga Plamenevskaya这项建议集中在三维接触拓扑和相关版本的纽结理论、Legendrian和横向纽结的研究。三维流形上的接触结构编码了亚拓扑学信息，而接触流形上的纽结(与接触平面相切或横向)具有许多附加性质。该项目的目标是更好地了解接触结构的各种不变量与勒让德结和横结点之间的关系，并找到新的应用。更具体地说，PI将研究Heegaard Floer同调中的接触、勒让德和横向不变量的各种版本；她建议通过相关的接触流形来研究纽结，例如通过手术获得的勒让德纽结和横向纽结的分支覆盖。该项目的一个相关部分是在量子代数(Khovanov和Khovanov-Rozansky同调)的纽结同调中开发类似的不变量。Heegaard Floer理论(通过全纯圆盘定义)和量子代数知识之间存在着接触几何之外的有趣关系；PI建议在存在接触结构的情况下研究这种关系。特别是，在她以前的工作中，PI提出了Khovanov同调中的一个横纽结不变量。Heegaard Floer和Khovanov同调之间的相互作用允许使用这个不变量来证明纽结的分枝双覆盖上的某些接触结构的紧性。这个项目的进一步进展可能会产生量子代数工具，用于在更一般的背景下研究接触结构，并将提高我们对性质截然不同的理论之间的关系的理解。拟议的研究是几何拓扑，这是一个研究不变维度的曲线空间(流形)的形状的数学领域。在维度3和维度4中，这可以被认为是对空间和时空结构的理解，并且特别有趣和重要。3维结的研究在拓扑学中起着重要的作用，并与其他科学有重要的联系(例如，DNA和某些蛋白质可以打结)。这项拟议的项目专注于研究存在接触结构的三维流形和节点，接触结构在某种程度上类似于物理学中的电场。(在历史上，接触结构的研究最初是由经典力学、光学和热力学推动的。)接触结构本身是重要的对象，但也编码了关于它们所在空间的有价值的信息。在她的研究中，PI计划使用几何和量子代数等不同数学分支的工具和想法。因此，该项目更广泛的目标包括更好地理解这些不同分支之间的关系，以及接触流形拓扑的新结果和应用。

AbstractAward: DMS-0805836Principal Investigator: Olga PlamenevskayaThis proposal focuses on contact topology in dimension 3 and therelated version of knot theory, the study of Legendrian andtransverse knots. Contact structures on 3-manifolds encode subtletopological information, and knots in contact manifolds (tangentor transverse to the contact planes) have many additionalproperties. The goal of this project is to better understand therelation between and find new applications of various invariantsof contact structures and Legendrian and transverse knots. Morespecifically, the PI will study various versions of the contact,Legendrian and transverse invariants in Heegaard Floer homology;she proposes to investigate knots via associated contactmanifolds, such as those obtained by surgery on Legendrian knotsand branched covers of transverse knots. A related part of theproject is to develop similar invariants in knot homologiesarising from quantum algebra (Khovanov and Khovanov-Rozanskyhomology). An intriguing relation between Heegaard Floer theory(defined via holomorphic disks) and the quantum algebraic knothomologies exists outside of contact geometry; the PI proposes tostudy this relation in presence of a contact structure. Inparticular, in her previous work the PI suggested a transverseknot invariant in Khovanov homology. The interplay betweenHeegaard Floer and Khovanov homologies allows to use thisinvariant to prove tightness of certain contact structures onbranched double covers of knots. Further progress on this projectcould yield quantum algebraic tools for studying contactstructures in a more general setting, and would improve ourunderstanding of the relation between theories of a verydifferent nature.The proposed research is on geometric topology, an area ofmathematics that studies shapes of curved spaces (manifolds) invarious dimensions. In dimensions 3 and 4, this can be thoughtof as understanding the structure of space and space-time, and isparticularly interesting and important. The study of knots indimension 3 plays a major role in topology and has importantconnection to other sciences (for example, DNA and certainproteins can be knotted). The proposed project focuses on studyof 3-manifolds and knots in presence of a contact structure, anobject somewhat analogous to an electric field in physics.(Historically, the study of contact structures was firstmotivated by classical mechanics, optics and thermodynamics.)Contact structures are important objects by themselves, but alsoencode valuable information about the space they live in. In herresearch, the PI plans to use tools and ideas from differentbranches of mathematics such as geometry and quantum algebra. Thebroader goals of the project thus include a better understandingof the relation between these different branches, as well as newresults in and applications of the topology of contact manifolds.