Variations of Right-Veering Open Books and Knot Positivity

该项目将推进几何和拓扑学中的两个重要学科——结点和链接理论（由多个组件组成的结点）和接触拓扑学的相互作用。该项目将推进辫状结构（看起来像实际辫状结构的链接）的研究，辫状结构在代数几何、算子代数、同伦理论、机器人、密码学、几何和拓扑学等领域有许多应用。该项目试图解决的基本问题是使用平面方式的图表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有潜力为美国STEM研究做出贡献的爱荷华州年轻人。项目目标是：(1)开发一种检测给定接触3流形的紧度或过扭度的图表方法，(2)将(1)的思想扩展到结理论（检测横向链接的不松动或松动），(3)从编织理论和接触几何的角度比较结和链接的各种正性，并找到它们的几何意义。使用的主要方法有Bennequin-Eliashberg不等式，Giroux对应，Birman和Menasco的辫状叶法，Ito和Kawamuro的开卷叶法。为了达到这一目标，我们研究了扭转-左转向映射类的性质以及3维和4维Bennequin-Eliashberg不等式的缺陷。项目范围为(1)研究Ito-Kawamuro引入的扭左转向与Wand定义的不一致性之间的关系，(2)继续研究扭左转向引起的横节深度，(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)紧密接触结构的研究，这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；(2)准正结在拓扑和代数几何中很重要。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.