On connected component decompositions of quandles

2017/04/25 by Iijima, Yusuke, Murao, Tomo · 1 citation
#57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1704.07689

Abstract

We give a formula of the connected component decomposition of the Alexander quandle: ℤ[t^±1]/(f₁(t),…, f_k(t))=\bigsqcup^a-1_i=0Orb(i), where a=gcd (f₁(1),…, f_k(1)). We show that the connected component Orb(i) is isomorphic to ℤ[t^±1]/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 connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.

Cited by

• Good involutions of twisted conjugation subquandles and Alexander quandles

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