Nilpotent quandles

TL;DR

This paper explores nilpotent quandles, revealing their structural properties, connections to nilpotent groups, and applications in link invariants, providing new characterizations and classifications.

Contribution

It introduces characterizations of generating sets, constructs free nilpotent quandles from free nilpotent groups, and links quandle properties to braid invariants.

Findings

Nilpotent quandles have the Hopf property.

A simple presentation of the associated group is provided.

Reduced fundamental quandles are equivalent to reduced peripheral systems.

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.