Subquandles of free quandles

Ivanov, Sergei O.; Kadantsev, Georgii; Kuznetsov, Kirill

doi:10.48550/arxiv.1904.06571

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Idempotents In Quandle Rings Of Free Products1

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2022

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“…Proof. An analogue of the Nielsen-Schreier theorem stating that every subquandle of a free quandle is free has been proved recently in [13]. Let F Q 2 = x ⋆ y be the free quandle of rank two.…”

Section: Idempotents In Quandle Rings Of Free Productsmentioning

confidence: 99%

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Idempotents, free products and quandle coverings

Elhamdadi¹,

Brandon²,

Singh³

et al. 2022

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In this paper we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of non-trivial involutory coverings over nice base quandles have infinitely many non-trivial idempotents and give their complete description. We show that the set of all these idempotents forms a quandle in itself. As an application, we deduce that the quandle ring of the knot quandle of a non-trivial long knot admit non-trivial idempotents. We consider free products of quandles and prove that integral quandle rings of free quandles have only trivial idempotents, giving an infinite family of quandles with this property. We also give a description of idempotents in quandle rings of unions and certain twisted unions of quandles.

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Section: Idempotents In Quandle Rings Of Free Productsmentioning

confidence: 99%

“…Adding(12) for all x ∈ X gives ε(v) = ε(v)ε(u). Similarly, adding(13) for all y ∈ Y gives ε(w) = ε(w)ε(u). If ε(u) = 0, then ε(v) = ε(w) = 0, and hence u = 0, a contradiction.…”

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confidence: 99%