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Variations of Right-Veering Open Books and Knot Positivity

右转向打开书籍和结积极性的变化

基本信息

批准号：
2005450
负责人：
Keiko Kawamuro
金额：
$ 22.19万
依托单位：
University of Iowa
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005450&HistoricalAwards=false
关键词：
Variations Right Veering Open Books

项目摘要

The project will advance interaction of the theory of knots and links (knots with several components) and contact topology, two significant subjects in Geometry and Topology. The project will progress the study of braids (links that look like actual braids) that has numerous applications to Algebraic Geometry, Operator Algebras, Homotopy Theory, Robotics, Cryptography, Geometry and Topology. The fundamental issue the project seeks to address is using diagrams on the plane way to detect whether contact structures are tight or overtwisted. Some parts of the project will involve research by graduate students. The Kids Topology Club will engage young Iowans who have potential to contribute to the STEM research of the US. Project goals are: (1) to develop a diagrammatic method to detect tightness or overtwistedness of a given contact 3-manifold, (2) to extend the idea of (1) to knot theory (detection of non-looseness or looseness of a transverse link), (3) to compare various positivities of knots and links from braid theory and contact geometry view points, and find their geometric meanings. Main methods used are the Bennequin-Eliashberg inequality, the Giroux correspondence, Birman and Menasco’s braid foliations, and Ito and Kawamuro’s open book foliations. To approach the goals, properties of twist-left-veering mapping classes and the defects of the 3- and 4-dimensional Bennequin-Eliashberg inequalities will be investigated. Scope of the project is (1) to study relation between twist-left-veering that Ito-Kawamuro introduce and inconsistency that Wand defined, (2) to continue investigating the depth of transverse knots via twist-left-veering, and (3) to study relation between the above-mentioned defects and Heegaard-Floer and Khovanov invariants. Potential contribution is to advance the study of (1) tight contact structures that is a central subject in contact geometry and has strong connection to symplectic geometry and algebraic geometry, and (2) quasipositive knots that are important in topology and algebraic geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目将推进节点和链接（具有多个组件的节点）和接触拓扑理论的相互作用，这是几何和拓扑中的两个重要主题。该项目将推进辫子（看起来像实际辫子的链接）的研究，该研究在代数几何，算子代数，同伦理论，机器人学，密码学，几何学和拓扑学中有许多应用。该项目寻求解决的基本问题是使用平面图来检测接触结构是否紧密或过度扭曲。该项目的某些部分将涉及研究生的研究。儿童拓扑俱乐部将吸引年轻的爱荷华州人谁有潜力为美国的STEM研究作出贡献。项目目标是：（2）将（1）的思想推广到纽结理论（检测横链的非松弛性或松弛性）;（3）从辫理论和接触几何的角度比较纽结和横链的各种正性，并找出它们的几何意义。使用的主要方法是Bennequin-Eliashberg不等式，Giroux对应，Birman和Menasco的辫子叶理，以及伊藤和川室的开卷叶理。为了达到这一目标，我们将研究扭曲左转向映射类的性质以及三维和四维Bennequin-Eliashberg不等式的缺陷。该项目的范围是（1）研究伊藤-川室介绍的扭曲左转向与Wand定义的不一致性之间的关系，（2）通过扭曲左转向继续研究横向结的深度，以及（3）研究上述缺陷与Heegaard-Floer和Khovanov不变量之间的关系。潜在的贡献是推进（1）紧接触结构的研究，紧接触结构是接触几何的中心课题，与辛几何和代数几何有很强的联系，以及（2）拓扑学和代数几何学中重要的准正结。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查进行评估，被认为值得支持的搜索.