Contact Topology, Knots, and Heegaard Floer Theory

摘要：项目负责人：Olga plamenevskaya本课题主要研究三维接触拓扑和相关版本的结理论，研究Legendrian结和横向结。3流形上的接触结构编码了微拓扑信息，并且接触流形中的结点（与接触平面横向的切线）具有许多附加性质。这个项目的目标是更好地理解接触结构和勒让德节和横节之间的关系，并找到新的应用。更具体地说，PI将研究Heegaard flower同调中的各种版本的接触不变量、Legendrian不变量和横向不变量；她建议通过相关的接触流形来研究结，例如那些通过手术获得的Legendrian结和横向结的分支覆盖。该项目的一个相关部分是开发由量子代数（Khovanov和Khovanov- rozanskyhomology）产生的结同调中的类似不变量。heeggaard flower理论（通过全纯磁盘定义）和量子代数知识论之间的有趣关系存在于接触几何之外；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同调中的横结不变量。heegaard Floer和Khovanov同调之间的相互作用允许使用该不变量来证明分支双结点盖上某些接触结构的紧密性。这个项目的进一步进展可能会产生量子代数工具，用于在更一般的环境中研究接触结构，并将提高我们对非常不同性质的理论之间关系的理解。提出的研究方向是几何拓扑学，这是一个研究不同维度弯曲空间（流形）形状的数学领域。在第三和第四维度，这可以被认为是理解空间和时空的结构，这是特别有趣和重要的。对3维结的研究在拓扑学中起着重要的作用，并且与其他科学（例如，DNA和某些蛋白质可以打结）有着重要的联系。这个项目的重点是研究接触结构下的3流形和结，接触结构有点类似于物理学中的电场。（历史上，接触结构的研究首先受到经典力学、光学和热力学的推动。）接触结构本身是重要的对象，但也编码了关于它们所处空间的有价值的信息。在她的研究中，PI计划使用来自不同数学分支的工具和思想，如几何和量子代数。因此，该项目的更广泛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.