Some Questions in Low-Dimensional and Contact Topology

本项目旨在开展接触和低维拓扑中几个相互关联的课题的研究。“接触结构”是流形上的额外几何结构（这些结构源自物理）；接触流形中接触结构和结点的研究借鉴了不同领域（几何、代数、组合学）的思想。该项目将有助于这些领域的发展，特别是流形的几何。该项目的主要部分是研究某些结的“右手性”和“左手性”性质。相关的思想经常出现在几何、代数和物理等更广泛的背景中。这个项目将关注两个不同的主题。第一个主题涉及3维的结理论和接触拓扑，涉及规范理论和组合性质的结不变量。该项目的目标是开发一类特殊的横向结（具有某些“正”性质的结）的性质和应用；PI希望找到相应辫的代数性质、结花和Khovanov同调的不变量的行为（包括PI在2006年引入的不变量）、连接结的辛曲面（协曲面）的性质以及它们的分支双盖的刚性之间的关系。本研究发展了一种新颖的方法（重点关注辫群排序和辫单性），但也延续了PI过去的研究。PI还计划继续研究高维接触拓扑中的柔性现象，并期望为这一快速发展的领域做出贡献。目标是更好地理解刚性/柔性接触流形的特性，开发3维现有结果的类似物。

This project aims to carry out research on several interrelated topics in contact and low-dimensional topology. A "contact structure'' is an extra geometric structure on a manifold (these structures originate from physics); the study of contact structures and knots in contact manifolds draws on ideas from different fields (geometry, algebra, combinatorics). This project will contribute to the development of these areas, especially to geometry of manifolds. The main part of the project concerns the investigation of "right-handed'' and "left-handed'' properties of certain knots. Related ideas often appear in the broader context of geometry, algebra, and physics.This project will focus on two different topics. The first topic concerns knot theory and contact topology in dimension 3, and is related to knot invariants of gauge-theoretic and combinatorial nature. The project's goal is to develop properties and applications for a special class of transverse knots (those with certain "positivity" properties); the PI expects to find relations between algebraic properties of corresponding braids, behavior of invariants from knot Floer and Khovanov homology (including an invariant introduced by the PI in 2006), properties of symplectic surfaces (cobordisms) connecting the knots, and rigidity of their branched double covers. This investigation develops a novel approach (with a focus on braid group orderings and the braid monodromy) but also continues the PI's past research. The PI also plans to continue studying flexible phenomena in higher-dimensional contact topology and expects to contribute to this rapidly developing area. The goal is to better understand properties of rigid/flexible contact manifolds, developing analogs of existing results in dimension 3.