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Nilpotent quandles

  • Jacques Darné
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Abstract

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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Cite

BibTeXRIS
BibTeX
@misc{darne2026nilpotent,
  title = {{Nilpotent quandles}},
  author = {Jacques Darné},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  eprint = {2205.02480},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-03659312v3},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Nilpotent quandles
AU  - Jacques Darné
PY  - 2026
DA  - 2026/10/02
PB  - HAL
UR  - https://hal.science/hal-03659312v3
AB  - 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.
ER  -