Combinatorial link homologies and their applications

本计画研究组合连结同源性的结构与应用。第一个主题是李代数的链同调和嵌入。已知李代数的某些嵌入导出相应多项式链不变量的状态和公式。主要研究者（PI）将研究如何将这些公式“提升”为这些链接多项式的简化。这样做的主要动机是一个潜在的明确分类的考夫曼多项式使用某些额外的结构上的Khovanov-Rozansky同源。第二个主题是链接图中的扭曲和Khovanov-Rozansky同调。Krasner简化了Khovanov-Rozansky双链扭曲链复合体，这导致了关于Khovanov-Rozansky同源性的新的拓扑和结构结果。PI将进一步研究这种简化链复合体的应用。这个项目的最后一个主题是生成树模型。PI将研究显式生成树模型的潜在应用的奇数Khovanov同源的结最近发现的罗伯茨，Jaeger和Manion。在20世纪90年代末，Khovanov指出了一个新的方向，在量子不变的方法，以低维拓扑。他的工作表明，而不是工作在一个线性空间或一个模块，应该工作在一个类别。通过这种方式，构造将导致比多项式不变量保留更多拓扑信息的同调不变量。这样的同调不变量被称为相应多项式不变量的重定向。霍瓦诺夫将琼斯多项式归类，并与罗赞斯基一起将HOMFLY-PT多项式归类。专家们已经使用这些简化来证明以前只能通过几何分析来获得的拓扑定理。其中最著名的是Rasmussen的组合证明Milnor猜想。该项目的目标是通过更好地理解现有的分类方法，构建新的分类方法并找到这些分类方法的新应用，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.