Quandles are the algebraic distillation of the Reidemeister moves on knot diagrams which renders them useful in the telling apart of knots. This leads us to study their structure. Right-invertibility (quandles 2nd axiom) allows us to regard them as sequences of permutations, the right translations. It has been interesting to look into finite quandles by way of the cycle decomposition these right translations may have. Hayashi's conjecture states that, for connected quandles, the length of any of the disjoint cycles divides the length of the longest one.

In this talk, we report on work done on this topic with Lages, Vojtechovsky and Singh.