Collaborative Research: Categorification of Link and 3-Manifold Invariants

该提案涉及环的代数定义的不变量和链余边论，称为环同调理论。在一个典型的例子中，这样的理论将二重同调群指定给一个环，并将群的同态指定给一个环的余边。作为这些理论的欧拉特征，链接的各种量子不变量得到了新的解释。链接同调理论可以被认为是四维拓扑量子场理论，仅限于三维空间中的链接和链接协边论。链同调的应用包括Milnor猜想的一个组合证明，纽结的thurston-Bennequin不变量的有效估计，以及许多纽结的双枝覆盖上不存在张叶。该提议的目的是加深对已知的链环同调理论及其相互关系的理解，包括Khovanov同调和纽结Floer同调之间的理解，发现链环同调在低维拓扑中的更多应用，发现归类于各种量子链环不变量的链环同调理论，将纽结Floer同调推广到纠缠，找到量子3-流形不变量的范畴，并探索链环同调与数学的各个分支的关系，包括同调代数和表示论。主要研究人员将把重点放在该理论的实验方面，包括编写有效的程序来计算链环同调。链环是位于我们生活的通常空间中的结圆的集合，而3-流形是在该空间上模拟的几何对象。研究链环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，代数就在区分链接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，也称为链接同调，是用一族代数对象替换已知的链接不变量的过程，这些代数对象显著增强了原始不变量。这个程序是最近由首席研究员开发的，他发现了几个链接的多项式不变量的分类。这些分类相对容易描述，但对于给定链接的计算却颇具挑战性。这个项目的目的是更好地理解现有的链接同调理论及其相互关系，并找到新的链接同调理论。主要研究人员还计划编写高效的程序来计算链接同源性。链接同源是一个年轻且发展迅速的领域。它位于三维和四维拓扑、辛拓扑、同调代数和表示论研究的十字路口。在可预见的未来，人们对连接同源性、其结构和应用的兴趣可能会继续增长。

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.