Collaborative Research: Categorification of Link and 3-Manifold Invariants

该提案涉及代数定义的链接和链接配边不变量，称为链接同源理论。在一个典型的例子中，这样一个理论指定双阶同调群的链接和一个同态群的链接配边。各种量子不变量的链接获得了新的解释，这些理论的欧拉特征。链同调理论可以看作是四维拓扑量子场论，仅限于三维空间中的链和链配边。链接同源性的应用包括组合证明的米尔诺猜想，有效的估计的Thurston-Bennequin不变量的结，和不存在的拉紧foliations对许多双分支覆盖的结。该提案的目的是加深对已知链接同调理论及其相互关系的理解，包括Khovanov同调和Knot Floer同调之间的关系，发现链接同调在低维拓扑中的更多应用，发现分类各种量子链接不变量的链接同调理论，将Knot Floer同调扩展到缠结，发现量子3流形不变量的简化，并探讨了链接同调与数学各分支的关系，包括同调代数和表示论。主要研究人员将重点放在理论的实验方面，包括编写高效的程序来计算链接homology.A链接是位于我们生活的通常空间中的打结圆的集合，3-流形是在该空间中建模的几何对象。研究链环和流形的主要问题之一是它们的分类，也就是说，找到将它们彼此区分开的方法。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，也称为链接同源，是一个用代数对象族替换已知链接不变量的过程，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，他发现了链接的几个多项式不变量的简化。这些映射相对容易描述，但对于给定链路的计算相当具有挑战性。该项目的目的是更好地了解现有的链接同源性理论及其相互关系，以及寻找新的。主要研究人员还计划编写计算连接同源性的有效程序。链接同源性是一个年轻和快速增长的领域。它是在3-和4-维拓扑，辛拓扑，同调代数和表示论的研究的十字路口。在可预见的将来，最近对连接同源性、其结构和应用的兴趣激增可能会继续下去。

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.