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On connected component decompositions of quandles. (English) Zbl 1467.57009
A quandle is an algebraic structure whose binary operation satisfies axioms coming from the three Reidemeister moves in knot theory. Orbits of a quandle \(X\) are obtained from the action of the inner automorphism group Inn\((X)\) on \(X\), (for more details see [M. Elhamdadi and S. Nelson, Quandles. An introduction to the algebra of knots. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1332.57007)]).
The Laurent polynomial ring \(\mathbb{Z}[t^{\pm 1}]\) is a quandle with the operation \(x*y=tx+(1-t)y\). Let \((f_1(t), \dots, f_n(t))\) be the ideal of \(\mathbb{Z}[t^{\pm 1}]\) generated by \(f_1(t), \dots, f_n(t)\). Then the quotient \(\mathbb{Z}[t^{\pm 1}]/ (f_1(t), \dots, f_n(t))\) becomes a quandle.
In this article, the authors determine the orbit decomposition of the Alexander quandle \(\mathbb{Z}[t^{\pm 1}]/ (f_1(t), \dots, f_n(t))\) (see Section 3). They also introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles (Section 4) and discuss the maximal connected sub-multiple conjugation quandle decomposition (Section 7).
57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)
Full Text: Euclid arXiv
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