On the Orbit Decomposition of Finite Quandles

Knot quandle decompositions

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the fundamental quandle of periodic links, composite knots and satellite knots.

On connected component decompositions of quandles

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a…

沿环面的纽结拟群分解

Knot quandle decomposition along a torus

本文研究了补包含不可压缩环面的纽结增广基本拟群结构，揭示了卫星纽结与其伴纽结及模式纽结基本拟群/群的关系，并提出了研究仿射拟群的代数方法。

The Orbit Group of a Quandle

We define the notion of the orbit group of a quandle via its connectivity and compute the orbit groups for some basic quandles. We also show that the orbit group counts the number of orbits of certain quandles.

拟群上良好对合的分解

Decompositions of good involutions on quandles

该研究探讨拟群两种基本构造下良好对合的行为，证明良好对合集的分解性质，并构造连通非对合对称拟群的例子作为对Ta猜想的反例。

重新定向淬链轨道

Reorienting quandle orbits

本文探讨了淬链轨道的重新定向概念，并指出该概念在媒介淬链中不适用。