Normal-form conjecture for elements of the monoid $\mathcal{O}_n$

Let $\mathcal{O}_n^{\alpha}$ and $\mathcal{O}_n^{\beta}$ denote the subcollections of elements represented using the corresponding $\alpha$- and $\beta$-letters. Let $u\in\mathcal{O}_n^{\alpha}$ and $v\in\mathcal{O}_n^{\beta}$ be in Jones normal form; let $i$ denote the first letter of $u$ and $j$ the last letter of $v$. A normal form of an element of $\mathcal{O}_n$ is one of the following:

$$uv,$$ $$\beta_iuv,$$ $$uv\alpha_j,$$ $$\beta_iuv\alpha_j.$$

Normal-form conjecture. In the second form, $u$ and $v$ satisfy restriction (a); in the third form, they satisfy restriction (b); and in the fourth form, they satisfy both restrictions, where (a) $u$ is as in Lemma

and $v$ is one of the possibilities listed in Corollary

, while (b) $u$ is one of the possibilities listed in Corollary

and $v$ is as in Corollary

. The displayed forms, with these restrictions, describe normal forms for elements of $\mathcal{O}_n$. The conjecture proposes a complete normal-form description of this monoid, building on the preceding structural lemmas and corollaries; the supplied text does not establish whether the description is complete or whether it has been proved or refuted.