Contact structures, open books, and connections between Heegaard Floer homology and the Khovanov-Rozansky link homology theories

Heegaard Floer同调是纽结，3-流形和光滑4-流形的不变量的集合，使用辛几何定义。它改变了景观的低维拓扑结构在过去的十年中，并已，例如，在许多最近的进展接触几何工具。该项目致力于更好地理解这个理论的内部工作原理及其与其他链接同源理论的关系，如Khovanov和Rozansky的HOMFLY多项式的分类。PI ™的广泛目标之一是使用这些关系来开发结、3-流形和接触结构的更可计算的不变量。另一个目标是使用Heegaard Floer同调来更好地理解开卷拓扑特征与接触结构几何性质之间的对应关系。具体的目标包括找到障碍物的接触结构具有支持属一，确定是否Legendrian手术保留紧密性，了解分数德恩扭曲系数和填充性之间的关系，并开发一个组合生成树模型结Floer同源性。PI计划使用后者来找到纽结Floer同调的公理化描述，部分是为了表明它与Kronheimer和Mrowka使用规范理论定义的瞬子和瞬子纽结同调一致。拟议的项目涉及研究3维和4维流形上的几何结构。理解这些空间和结构是理解我们宏观宇宙形状的核心，PI打算研究的一些理论在物理学中有应用和起源。特别是接触结构，可以作为研究这些空间的探针，并且在经典力学，热力学，动力学系统和液晶研究中也很重要。PI提案的另一个目标是开发理解结的新方法。结在像DNA这样的分子中自然产生。例如，最近的工作已经应用Floer同源性来确定体内的各种酶如何改变DNA的打结。了解这些机制对于开发某些药物非常重要。

Heegaard Floer homology is a collection of invariants of knots, 3-manifolds, and smooth 4-manifolds, defined using symplectic geometry. It has changed the landscape of low-dimensional topology over the last decade, and has, for example, been instrumental in many of the recent advances in contact geometry. This project is devoted to better understanding the internal workings of this theory and its relationships to other link homology theories like Khovanov and Rozansky¢s categorification of the HOMFLY polynomial. One of the PI¢s broad goals is to use these relationships to develop more computable invariants of knots, 3-manifolds, and contact structures. Another goal involves using Heegaard Floer homology to better understand the correspondence between topological features of open books and geometric properties of contact structures. Specific aims include finding obstructions to a contact structure having support genus one, determining whether Legendrian surgery preserves tightness, understanding the relationship between fractional Dehn twist coefficient and fillability, and developing a combinatorial spanning tree model for knot Floer homology. The PI plans to use the latter to find an axiomatic description of knot Floer homology, in part to show that it agrees with the monopole and instanton knot homologies defined by Kronheimer and Mrowka using gauge theory.The proposed project involves studying geometric structures on 3- and 4-dimensional manifolds. Understanding these spaces and structures is central to understanding the shape of our macroscopic universe, and some of the theories the PI intends to study have applications and origins in physics. Contact structures, in particular, can serve as a probe with which to study these spaces, and are also important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. Another goal of the PI¢s proposal is to develop new methods for understanding knots. Knots arise naturally in molecules like DNA. Recent work, for example, has applied Floer homology to determine how various enzymes in the body alter the knottedness of DNA. Understanding these mechanisms is important for developing certain drugs.