多元Quandle作为群胚不变量
Multivariate Quandles as Groupoid Invariants
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中文总结 AI 辅助
本文提出基于群胚的多元Alexander quandle构造框架，实现Alexander模的oid化，用于枚举新quandle类并寻找链环新着色不变量。
中文摘要 AI 辅助
我们引入了一个新的结构框架，用于基于群胚${\cal G}$构造多元Alexander quandle，该群胚由覆盖群的去环群胚的不相交并集组成。与群环${\mathbb{Z}}[{\mathbb{Z}}]$上的标准Alexander模相比，多元版本涉及群胚环${\mathbb{Z}}[{\cal G}]$上的模。我们以${\mathbb{Z}}$-Algebroid给出了一个明显的范畴论描述，其函子模理论产生了多元Alexander模的等价定义。利用元素范畴构造，我们从该群胚框架定义了多元Alexander quandle。随之而来的是三类多元quandle运算。与早期方法不同，我们从显式的链环依赖性中抽象出代数结构。因此，我们有了一个结构框架，用于：（i）枚举新的Alexander quandle类，以及（ii）寻找链环的新着色不变量。我们证明了通过多元quandle运算实现的${\mathbb{Z}}$-模之间的映射实现了quandle的箭图表示，这使得Alexander模的“oid化”变得明显。基于群胚的oid化Alexander模为组合单变量quandle以获得新的多元quandle提供了构造性框架。最后，我们评论了基于群胚的公式（包括基本群胚）产生新的链环不变量的可能性。
英文摘要
We introduce a new structural framework for constructing multivariate Alexander quandles based on a groupoid ${\cal G}$, composed of a disjoint union of delooping groupoids of deck groups. Compared to standard Alexander modules over group rings ${\mathbb{Z}}[{\mathbb{Z}}]$, the multivariate version involves modules over groupoid rings ${\mathbb{Z}}[{\cal G}]$. A manifestly categorical description is given in terms of ${\mathbb{Z}}$-Algebroids, whose functorial module theory yields an equivalent definition of multivariate Alexander modules. Using the category of elements construction, we define multivariate Alexander quandles from this groupoid framework. Three classes of multivariate quandle operations follow. In contrast to earlier approaches, we abstract the algebraic structure from explicit link-dependence. As a consequence, we have a structural framework for: (i) enumerating new classes of Alexander quandles, and (ii) finding new coloring invariants of links. We show that maps between ${\mathbb{Z}}$-modules, facilitated via multivariate quandle operations, realize a quiver presentation of quandles, which makes manifest the oidification of Alexander modules. Oidified Alexander modules, based on groupoids, provide a constructive framework for composing univariate quandles to obtain new multivariate ones. Finally, we comment on the possibility of new link invariants coming from groupoid-based formulations, including the fundamental groupoid.
发表机构
- Wolfram Institute for Computational Foundations of Science（Wolfram计算科学基础研究所）
- University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
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