Idempotents, free products and quandle coverings

Abstract

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.

Enhancements of link colorings via idempotents of quandle rings

“…In Section 6, we present computer assisted calculations of all idempotents for integral as well as mod 2 quandle rings of all quandles of order less than six, and also determine quandles for which the set of all idempotents is itself a quandle. The data also supports our conjecture [8,Conjecture 3.10] about triviality of idempotents of integral quandle rings of finite latin quandles, and also suggests that any non-zero idempotent of an integral quandle ring has augmentation value 1 (Conjecture 6.2). Finally, in Section 7, we discuss Peirce spectra for complex quandle algebras of quandles of order three.…”

“…) is a trivial quandle with respect to the ring multiplication (see also [8,Corollary 4.12]). More generally, the following is a consequence of [8, Theorem 4.5].…”

“…g(e w ) g(e x ) g(e w )g(e x ) = g(e w )g(e x ) g(e w ), (8) g(e x ) g(e w )g(e x ) g(e w ) = g(e x ) g(e w )g(e x ) .…”

“…In contrast, idempotents are the most natural objects in quandle rings since each quandle element is, by definition, an idempotent of the quandle ring. Exploiting the idea of a quandle covering as developed in [4,5], the complete set of idempotents of the quandle ring of an involutory covering, when the quandle ring of the base has only trivial idempotents, has been determined in [8]. Interestingly, the set of all idempotents in this case is itself a quandle with respect to the ring multiplication.…”

“…We shall see that this phenomenon occurs more often in mod 2 quandle rings. In addition, integral quandle rings of free quandles have been shown to admit only trivial idempotents [8], which gives an infinite family of quandles with this property.…”