Low-Dimensional Topology from a "Super" View Point

该项目有三个组成部分，都涉及低维拓扑和量子代数之间的相互作用。提出的研究的第一个主要主题是多变量链接不变量。首席研究者和Patureau-Mirand已经证明了李超代数自然地产生了多变量链接不变量。该项目继续这项工作，目的是进一步了解这些不变量以及它们与其他链接不变量的关系。这一初步工作表明，这些多变量链接不变量与Volume猜想有关。拟议的研究还将集中在由特定李超代数产生的量子不变量的分类上。这可以给出亚历山大多项式的拓扑分类，在Bar-Natan对Khovanov同调的拓扑解释的意义上。这项研究的最后一个主题是构造一个与李超代数相关的3-流形不变量。结出现在许多地方，从系鞋带到弦理论和蛋白质折叠等日常用品。在数学中，节点和更一般的链接是拓扑学中的基本基本对象。自19世纪以来，人们一直在研究绳结。最近，随着量子场论的观点进入低维拓扑学，20世纪末出现了一场节点和链接理论的革命。特别是，与李代数相关的量子群理论在低维拓扑中得到了广泛而卓有成效的应用。与李超代数相关的量子群理论及其应用还不是很发达。李超代数理论具有李代数理论所没有的特殊性质。这个方案的重点是利用这些独特的性质来获得关于低维拓扑中的已知问题和对象的新信息。

The project has three components, all concerning interactions between low-dimensional topology and quantum algebra. The first main theme of the proposed research is multivariable link invariants. The Principal Investigator and Patureau-Mirand have shown that Lie superalgebras naturally give rise to multivariable link invariants. The project continues this work with an aim of further understanding these invariants and how they relate to other link invariants. This initial work suggests that these multivariable link invariants are related to the Volume Conjecture. The proposed research will also focus on the categorification of a quantum invariant arising from a particular Lie superalgebra. This could give a topological categorification of the Alexander polynomial, in the sense of Bar-Natan's topological interpretation of Khovanov homology. A final theme in the proposed research is the construction of a 3-manifold invariant associated to a Lie superalgebra.Knots appear in many places ranging from everyday items such as tied shoe laces to string theory and protein folding. In mathematics, knots and more generally links are fundamental basic objects in topology. Knots have been studied since the 19th century. More recently, with the quantum field theory point of view moving into low-dimensional topology, the late 20th century saw the emergence of a revolution in the theory of knots and links. In particular, the theory of quantum groups associated to Lie algebras has been widely and productively used in low-dimensional topology. Not as well developed is the theory of quantum groups associated to Lie superalgebras and its applications. The theory of Lie superalgebras has particular properties that do not arise in the theory of Lie algebras. The focus of this proposal is to use these unique properties to gain new information about well known problems and objects in low-dimensional topology.