Idempotents, free products and quandle coverings

TL;DR

This paper explores the structure of idempotents in quandle rings, revealing their connection to quandle coverings and free products, with implications for knot theory and algebraic structures.

Contribution

It provides a complete description of non-trivial idempotents in quandle rings related to coverings and shows that free quandles have only trivial idempotents, advancing understanding of quandle algebraic properties.

Findings

Integral quandle rings of finite type coverings have infinitely many non-trivial idempotents.

Quandle rings of free quandles have only trivial idempotents.

The set of idempotents in certain quandle rings forms a quandle itself.

Abstract

In this paper, we investigate idempotents in quandle rings and relate them with quandle coverings. We prove that integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit 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.