Classification of idempotents in the Okada monoid

Classify all idempotent elements of the Okada monoid, including in the unlabelled Jones monoid case, and determine a general characterization extending beyond the involutive idempotents.

Background

The paper proves that every involutive Okada monoid element is idempotent, but emphasizes that involutive elements do not account for all idempotents. It gives examples in low ranks showing that the idempotent structure is substantially larger and more complicated than the involutive subclass.

The authors report computational counts of idempotents for ranks 0 through 10 and note that the classification problem remains difficult even for the unlabelled Jones monoid. Thus the unresolved task is to characterize precisely which labelled or unlabelled diagram elements are idempotent.

References

We have not yet managed to fully understand which elements of $\Okada$ are itempotents. This problem seems to be non-trivial even in the unlabelled case, i.e. for the Jones monoid~(see).

— Diagrammatic Okada monoid and cellularity of the Okada algebra  (2609.01440 - Hivert et al., 1 Sep 2026) in Section 4, subsection “Aperiodicity of the Okada monoid,” remark following Corollary 4.?, before the discussion of regularity