Combinatorial link homologies and their applications

组合链接同源性及其应用

• 批准号: 1205879
• 负责人: Hao Wu
• 金额: $ 12.57万
• 依托单位: George Washington University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2017-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1205879&HistoricalAwards=false
• 关键词: Combinatorial; link; homologies; their; applications

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.

该项目涉及组合链接同源性的结构和应用。拟议研究的第一个主题是李代数的链接同调和嵌入。众所周知，李代数的某些嵌入会导出相应多项式链接不变量的状态和公式。首席研究员（PI）将研究如何将这些公式“提升”到这些链接多项式的分类。这样做的主要动机是使用 Khovanov-Rozansky 同调上的某些附加结构对考夫曼多项式进行潜在的显式分类。第二个主题是链接图的扭曲和 Khovanov-Rozansky 同源性。克拉斯纳简化了两链扭曲的霍瓦诺夫-罗赞斯基链复合体，这导致了关于霍瓦诺夫-罗赞斯基同源性的新拓扑和结构结果。 PI 将进一步研究这种简化链复合体的应用。该项目的最终主题是生成树模型。 PI 将研究 Roberts、Jaeger 和 Manion 最近发现的奇特霍瓦诺夫结同调的显式生成树模型的潜在应用。在 20 世纪 90 年代末，霍瓦诺夫指出了低维拓扑的量子不变方法的新方向。他的工作表明，人们不应该在线性空间或模块内工作，而应该在类别内工作。这样，构造将产生同调不变量，它比多项式不变量保留更多的拓扑信息。这种同调不变量称为相应多项式不变量的分类。 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。专家们利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。该项目的目标是通过更好地理解现有分类、构建​​新分类并寻找这些分类的新应用，进一步扩大结理论分类方法的范围。