增广Racks的扩张与曲面带余圈不变量
Extensions of Augmented Racks and Surface Ribbon Cocycle Invariants
摘要：
格是一个具有右可逆和自分配的二元运算的集合，其性质分别对应于Reidemeister移动II和III。如果操作是由群作用写入的，则机架被称为{ \it增强机架}。Racks及其上同调理论已被广泛用于纽结和纽结曲面不变量。类似于群上同调，2-上圈与扩张有关，一个自然的问题是刻画增广架的扩张本身就是增广架。在本文中，我们刻画了这样的扩展方面，我们称之为{ \it补充和添加剂}上同调的机架。同时扩展的机架和组被认为是，其中相应的$2$ -cocycles相关的通过一定的公式。进一步，我们构造了嵌入$3$ -空间中的带形边界紧致可定向曲面的染色和上循环不变量。
英文摘要：
A rack is a set with a binary operation that is right-invertible and self-distributive, properties diagrammatically corresponding to Reidemeister moves II and III, respectively. A rack is said to be an {\it augmented rack} if the operation is written by a group action. Racks and their cohomology theories have been extensively used for knot and knotted surface invariants. Similarly to group cohomology, rack 2-cocycles relate to extensions, and a natural question that arises is to characterize the extensions of augmented racks that are themselves augmented racks. In this paper, we characterize such extensions in terms of what we call {\it fibrant and additive} cohomology of racks. Simultaneous extensions of racks and groups are considered, where the respective $2$-cocycles are related through a certain formula. Furthermore, we construct coloring and cocycle invariants for compact orientable surfaces with boundary in ribbon forms embedded in $3$-space.
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