Enhancements of link colorings via idempotents of quandle rings

Abstract

We show that quandle rings and their idempotents lead to proper enhancements of the well-known quandle coloring invariant of links in the 3-space. We give explicit examples to show that the new invariants are also stronger than the $\Hom$ quandle invariant when the coloring quandles are medial. We provide computer assisted computations of idempotents for all quandles of order less than six, and also determine the ones for which the set of all idempotents is itself a quandle. The data supports our conjecture about triviality of idempotents of integral quandle rings of finite latin quandles. We also determine Peirce spectra for complex quandle algebras of some small order quandles.