Collaborative Research: Algebraic Structures and Cohomology Theories Associated to Knottings

主要研究人员和合作者利用称为纽结和曲面图的自分配运算和它们的染色，以状态和的形式定义了三维空间中的纽结和四维空间中的纽结曲面的新的状态和不变量。状态和的权重由Quandle上同调理论得到。人们已经发现了在结和表面的各种性质上的许多应用。该项目研究了Quandles、Lie代数、余代数、交叉模及其上同调理论之间的关系，以开发诸如流形不变量等应用。它还建议使用几何和图解方法来分析特定的范畴、量子群和上同调理论。结是位于空间中的圆。四维空间中的曲面也可以打结。纽结理论研究这种打结的圆和曲面，并为DNA理论、分子构型和物理学提供了模型和应用。在一张纸上画的纽结图，以及从图中很容易计算的数值，在纽结理论中得到了广泛的应用。主要研究人员和他们的合作者从纽结图开发了代数系统，这些系统给出了对纽结的视觉表示的近距离反映。这些图的代数和相关版本简明地编码了结点和物理系统之间的深层联系。目前的项目开发了图的代数系统和其他已建立的代数系统(李代数和交叉模)之间的新联系，这些代数系统与物理学中的标准模型密切相关。这些技术还将应用于分类-一个用如何识别的指令取代身份的过程。

New state-sum invariants for knots in 3-dimensional space and knotted surfaces in 4-dimensional space were defined, in a state-sum form, by the principal investigator and collaborators, using self-distributive operations called quandles and their colorings of knot and surface diagrams. The weights of the state-sum are derived from quandle cohomology theories. A number of applications to various properties of knots and surfaces have been discovered. The project investigates relationships among quandles, Lie algebras, coalgebras, crossed modules and their cohomology theories in order to develop applications such as manifold invariants. It also proposes to use geometric and diagrammatic methods to analyse specific categorifications, quantum groups, and cohomology theories.A knot is a circle situated in space. Surfaces in four-dimensional space can also be knotted. Knot theory studies such knotted circles and surfaces, and has provided models and applications to DNA theory, molecular configurations, and physics. Knot diagrams drawn on a piece of paper, and numerical quantities that are easily computable from diagrams, have been extensively used in knot theory. The principal investigators and their collaborators have developed algebraic systems from the knot diagrams that give a close reflection of the visual representations of knots. The algebra of these diagrams and related versions concisely encode deep connections among knots and physical systems. The current project develops new connections between the algebraic system of diagrams and other established algebraic systems (Lie algebras and crossed modules) that are closely associated with the standard model in physics. The techniques will also be applied in the context of categorification --- a process by which identity is replaced by an instruction of how to identify.