Collaborative Research: Categorification of Link and 3-Manifold Invariants

该建议处理代数上定义的连杆不变量和连杆共轭，称为连杆同调理论。在一个典型的例子中，该理论将群的同态赋给一个环，将群的同态赋给一个环协。链接的各种量子不变量作为这些理论的欧拉特性获得了一种新的解释。链接同调理论可以被认为是四维拓扑量子场论，局限于三维空间中的链接和链接协同。连杆同调的应用包括Milnor猜想的组合证明，结的Thurston-Bennequin不变量的有效估计，以及在许多结的双分支覆盖上紧叶的不存在性。本课题旨在深化对已知连杆同调理论及其相互关系的理解，包括Khovanov同调与knot Floer同调之间的关系，寻找连杆同调在低维拓扑中的更多应用，发现对各种量子连杆不变量进行分类的连杆同调理论，将knot Floer同调扩展到纠缠，寻找量子3流形不变量的分类，并探讨了连杆同调与数学各分支的关系，包括同调代数和表示理论。主要研究人员将把重点放在理论的实验方面，包括编写有效的程序来计算链接同源性。链接是位于我们生活的通常空间中的打结的圆的集合，而3流形是在该空间中建模的几何对象。研究链接和流形的主要问题之一是它们的分类，即找到区分它们的方法。从一开始，代数就在区分连杆和流形方面发挥了重要作用，因为代数对象通常比几何对象更容易相互比较。分类，也被称为链接同调，是用一组代数对象取代已知的链接不变量的过程，这些代数对象显著增强了原来的不变量。这个程序是最近由首席研究员开发的，他发现了几个多项式不变量的链接的分类。这些分类相对容易描述，但对于给定的链接计算起来却相当困难。该项目的目的是为了更好地了解现有的链接同源理论及其相互关系，并寻找新的理论。主要研究人员还计划编写有效的程序来计算链路同源性。链接同源是一个年轻而迅速发展的领域。它位于三维和四维拓扑、辛拓扑、同调代数和表示理论研究的十字路口。最近对链接同源性及其结构和应用的兴趣激增，在可预见的未来可能会持续下去。

The proposal deals with algebraically defined invariants of links and link cobordisms, known as link homology theories. In a typical instance, such a theory assigns bigraded homology groups to a link and a homomorphism of groups to a link cobordism. Various quantum invariants of links gain a novel interpretation as the Euler characteristics of these theories. Link homology theories can be though of as four-dimensional topological quantum field theories, restricted to links in the 3-space and link cobordisms. Applications of link homology include a combinatorial proof of the Milnor conjecture, efficient estimates of the Thurston-Bennequin invariant of knots, and nonexistence of taut foliations on many double branched covers of knots. The proposal's aim is to deepen understanding of known link homology theories and their interrelations, including the ones between Khovanov homology and knot Floer homology, find more applications of link homology to low-dimensional topology, discover link homology theories categorifying a variety of quantum link invariants, extend knot Floer homology to tangles, find categorifications of quantum 3-manifold invariants, and explore the relations of link homology to various branches of mathematics, including homological algebra and representation theory. The principal investigators will place a significant emphasis on experimental aspects of the theory, including writing efficient programs to compute link homology.A link is a collection of knotted circles located in the usual space that we live in, and 3-manifolds are geometric object modeled on that space. One of the main problems in studying links and manifolds is their classification, that is, finding methods to tell them apart from each other. From the very beginning, algebra played an important role in distinguishing links and manifolds, since algebraic objects are in general easier to compare to each other than geometric ones. Categorification, also known as link homology, is a procedure of replacing a known link invariant with a family of algebraic objects that significantly enhance the original invariant. This procedure was recently developed by the lead principal investigator, who found categorifications of several polynomial invariants of links. These categorifications are relatively easy to describe, but rather challenging to compute for a given link. The aim of this project is to better understand existing link homology theories and their interrelations, as well as to find new ones. The principal investigators also plan to write efficient programs for computing link homology. Link homology is a young and quickly growing field. It lies on the crossroads of research in 3- and 4-dimensional topology, symplectic topology, homological algebra, and representation theory. The recent explosion of interest in link homology, its structure and applications, is likely to continue in the foreseeable future.