Some Questions in Low-Dimensional and Contact Topology

本项目旨在对接触和低维拓扑中几个相互关联的课题进行研究。“接触结构”是流形上的额外几何结构（这些结构起源于物理学）;接触结构和接触流形中的结的研究借鉴了不同领域（几何学，代数学，组合学）的思想。这个项目将有助于这些领域的发展，特别是流形的几何。该项目的主要部分涉及的“右手”和“左手”的某些节属性的调查。相关的概念经常出现在几何、代数和物理的大背景下。这个专题将集中在两个不同的主题上。第一个主题涉及结理论和接触拓扑在3维，并涉及到结不变量的规范理论和组合性质。该项目的目标是开发一种特殊类型的横向结的性能和应用（具有某些“积极性”属性）; PI希望找到相应辫子的代数性质之间的关系，来自结Floer和Khovanov同调的不变量的行为（包括PI在2006年引入的不变量），连接节点的辛曲面（协边）的性质，以及它们的分支双覆盖的刚性。这项研究开发了一种新的方法（重点是辫子群排序和辫子单值），但也继续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。我们的目标是更好地了解刚性/柔性接触流形的属性，在三维现有的结果开发类似物。

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.