Abelian quandles and quandles with abelian structure group

Lebed, Victoria; Mortier, Arnaud

doi:10.1016/j.jpaa.2020.106474

Cited by 6 publications

(12 citation statements)

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“…In this section, we apply the results of the previous section to classify 2-nilpotent quandles, and to compute their enveloping groups. 2-nilpotent quandles are the one called abelian in [LM21], and we recover their classification results with a slightly different point of view. In particular, we recover the calculations of [Eis14, Prop.…”

Section: -Nilpotent Quandlessupporting

confidence: 52%

“…A quandle Q is 2-nilpotent when Inn(Q) is abelian, that is, when x (y (−)) = y (x (−)) for all x, y ∈ Q. We call such quandles abelian, following [LM21], whose classification we recover in §5.3.…”

Section: Definitionmentioning

confidence: 99%

“…If we want to do computations, we need a precise way of describing the H i . This is the role played by parameters from [LM21]. Before introducing them, let us remark that H i must contain e i , so we only need to describe its image H i in Z n /e i ∼ = Z n−1 .…”

Section: Finite-type 2-nilpotent Quandlesmentioning

confidence: 99%

“…3.6], where it was asked if one can classify quandles having a nilpotent enveloping group. In particular, we recover the classification of abelian quandles from [LM21].…”

Section: Introductionmentioning

confidence: 99%

“…We use it to find our presentation of enveloping groups of nilpotent quandles, and to give a useful injectivity criterion. We then turn our attention to the particular case of 2-nilpotent quandles (called abelian in [LM21]), whose study is the subject of §5. Finally, the last two sections are more closely related to the theory of knots and links.…”

Section: Introductionmentioning

confidence: 99%

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Nilpotent quandles

Darné¹

2022

Preprint

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A nilpotent quandle is a quandle whose inner automorphism group is nilpotent. Such quandles have been called reductive in previous works, but it turns out that their behaviour is in fact very close to nilpotency for groups. In particular, we show that it is easy to characterise generating sets of such quandles, and that they have the Hopf property. We also show how to construct free nilpotent quandles from free nilpotent groups. We then use the properties of nilpotent quandles to describe a simple presentation of their associated group, and we use this to recover the classification of abelian quandles by Lebed and Mortier [LM21]. We also study reduced quandles, and we show that the reduced fundamental quandle is equivalent, as an invariant of links, to the reduced peripheral system, sharpening a previous result of Hughes [Hug11]. Finally, we give a characterisation of nilpotency in terms of the associated invariants of braids.

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Section: -Nilpotent Quandlessupporting

confidence: 52%

Section: Definitionmentioning

confidence: 99%

Section: Finite-type 2-nilpotent Quandlesmentioning

confidence: 99%

“…3.6], where it was asked if one can classify quandles having a nilpotent enveloping group. In particular, we recover the classification of abelian quandles from [LM21].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations