Tokyo Journal of Mathematics

On Connected Component Decompositions of Quandles

Yusuke IIJIMA and Tomo MURAO

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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 connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.

Article information

Tokyo J. Math., Volume 42, Number 1 (2019), 63-82.

First available in Project Euclid: 18 July 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds


IIJIMA, Yusuke; MURAO, Tomo. On Connected Component Decompositions of Quandles. Tokyo J. Math. 42 (2019), no. 1, 63--82.

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