On torus knot groups and a submonoid of the braid group

The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $\sigma_1$ and $\sigma_1 \sigma_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of Garside monoids generalizing this exotic Garside structure, i.e., such that $M_2$ is isomorphic to the above monoid. The corresponding Garside group $G(M_n)$ is isomorphic to the $(n,n+1)$-torus knot group-which is isomorphic to $\mathcal{B}_3$ for $n=2$ and to the braid group of the exceptional complex reflection group $G_{12}$ for $n=3$. This yields a new Garside structure on $(n,n+1)$-torus knot groups, which already admit several distinct Garside structures. The $(n,n+1)$-torus knot group is an extension of $\mathcal{B}_{n+1}$, and the Garside monoid $M_n$ surjects onto the submonoid $\Sigma_n$ of $\mathcal{B}_{n+1}$ generated by $\sigma_1, \sigma_1 \sigma_2, \dots, \sigma_1 \sigma_2\cdots \sigma_n$, which is not a Garside monoid when $n>2$. Using a new presentation of $\mathcal{B}_{n+1}$ that is similar to the presentation of $G(M_n)$, we nevertheless check that $\Sigma_n$ is an Ore monoid with group of fractions isomorphic to $\mathcal{B}_{n+1}$, and give a conjectural presentation of it, similar to the defining presentation of $M_n$. This partially answers a question of Dehornoy-Digne-Godelle-Krammer-Michel.