Combinatorial link homologies and their applications

本课题研究的是组合链同调的结构和应用。研究的第一个主题是李代数的链同调和嵌入。众所周知，李代数的某些嵌入导出了相应多项式链接不变量的状态和公式。首席调查员(PI)将研究如何将这些公式“提升”到这些链接多项式的范畴。这样做的主要动机是使用Khovanov-Rozansky同调上的某些附加结构对Kauffman多项式进行潜在的显式分类。第二个主题是链接图中的扭曲和Khovanov-Rozansky同调。Krasner将Khovanov-Rozansky链复合体简化为两链扭曲，这导致了关于Khovanov-Rozansky同源的新的拓扑和结构结果。PI将进一步研究这种简化的链状络合物的应用。本项目的最后一个主题是生成树模型。PI将研究由Roberts，Jaeger和Manion最近发现的纽结的奇Khovanov同调的显式生成树模型的潜在应用。在20世纪90年代末，Khovanov指出了低维拓扑的量子不变量方法的一个新方向。他的研究表明，人们不应该在线性空间或模块内工作，而应该在一个范畴内工作。这样，构造的同调不变量将比多项式不变量保留更多的拓扑信息。这样的同调不变量称为相应多项式不变量的范畴。Khovanov对Jones多项式进行了分类，并与Rozansky一起对HOMFLY-PT多项式进行了分类。专家们已经使用这些分类来证明以前只能通过几何分析才能获得的拓扑定理。其中，最著名的是拉斯穆森的组合证明米尔诺猜想。本项目的目标是通过更好地理解现有的分类，构建新的分类，并找到这些分类的新应用，进一步扩大纽结理论的分类方法的范围。

This project deals with structures and applications of combinatorial link homologies. The first theme of the proposed research is link homologies and embeddings of Lie algebras. It is known that certain embeddings of Lie algebras induce state sum formulas for the corresponding polynomial link invariants. The Principal Investigator (PI) will study how to "lift" such formulas to the categorifications of these link polynomials. The main motivation for this is a potential explicit categorification of the Kauffman polynomial using certain additional structures on the Khovanov-Rozansky homology. The second theme is twistings in link diagrams and the Khovanov-Rozansky homology. Krasner simplified the Khovanov-Rozansky chain complex of a two-strand twisting, which has led to new topological and structural results about the Khovanov-Rozansky homology. The PI will further study applications of this simplified chain complex. The final theme of this project is the spanning tree model. The PI will study potential applications of the explicit spanning tree model for the odd Khovanov homology of knots recently discovered by Roberts, Jaeger and Manion.In the late 1990s, Khovanov pointed to a new direction in the quantum invariant approach to low-dimensional topology. His work showed that, instead of working inside a linear space or a module, one should work inside a category. This way, the construction will lead to homological invariants which retain more topological information than the polynomial invariants. Such homological invariants are called categorifications of the corresponding polynomial invariants. Khovanov categorified the Jones polynomial and, with Rozansky, the HOMFLY-PT polynomial. Experts have used these categorifications to prove topological theorems that were only accessible by geometric analysis before. Among these, the best known is Rasmussen's combinatorial proof Milnor's Conjecture. The goal of this project is to further expand the scope of the categorification approach to knot theory by better understanding existing categorifications, constructing new categorifications and finding new applications of these categorifications.