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

Heegaard Floer同调是用辛几何定义的纽结、3-流形和光滑4-流形的不变量的集合。在过去的十年里，它改变了低维拓扑的格局，例如，在接触几何的许多最新进展中发挥了重要作用。这个项目致力于更好地理解这一理论的内部工作原理及其与其他链环同调理论的关系，如霍万诺夫和罗赞斯基，S对HOMFLY多项式的分类。S的主要目标之一是利用这些关系来开发纽结、3-流形和接触结构的更可计算的不变量。另一个目标是利用Heegaard Floer同调来更好地理解开卷的拓扑特征和接触结构的几何性质之间的对应关系。具体目标包括寻找支承亏格为1的接触结构的障碍，确定Legendrian手术是否保持紧密性，了解分数Dehn扭转系数与可填充性之间的关系，以及开发纽结Floer同调的组合生成树模型。PI计划使用后者来寻找纽结Floer同调的公理描述，部分地是为了证明它与Kronheimer和Mrowka使用规范理论定义的单极子和瞬子纽结同调是一致的。理解这些空间和结构是理解我们宏观宇宙形状的核心，而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，在经典力学、热力学、动力学系统和液晶研究中也是重要的。派-S提议的另一个目标是发展理解结的新方法。像DNA这样的分子中自然会打结。例如，最近的研究应用Floer同源性来确定体内的各种酶如何改变DNA的结节。了解这些机制对于开发某些药物很重要。

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.