Enhancements of link colorings via idempotents of quandle rings

TL;DR

This paper introduces enhanced link invariants derived from quandle rings and their idempotents, demonstrating their strength over existing invariants through explicit examples and computational analysis.

Contribution

It develops new link invariants using quandle ring idempotents, compares their strength to existing invariants, and provides computational data on idempotents for small quandles.

Findings

New invariants are stronger than $ ext{Hom}$ quandle invariants for medial quandles.

Explicit computations of idempotents for all quandles of order less than six.

Conjecture on triviality of idempotents in integral quandle rings of finite latin quandles.

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.