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 ${(M_n)}_{n\ge 1}$ 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 ${\mathrm{\Sigma }}_n$ 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 ${\mathrm{\Sigma }}_n$ 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.

3股辫子群的亚类恐龙 ${\mathcal{乙}}_3$ 由...产生 ${\sigma }_{\mathrm{1个}}$ 和 ${\sigma }_{\mathrm{1个}}{\sigma }_{\mathrm{2个}}$ 已知会产生一种奇异的Garside结构 ${\mathcal{乙}}_3$。我们介绍和研究一个无限的家庭${（{\mathrm{中号}}_{ñ}）}_{ñ\ge \mathrm{1个}}$加塞德类群概括这种外来Garside的结构，即，使得${\mathrm{中号}}_{\mathrm{2个}}$与上述类半同构同构。对应的Garside组$G（{\mathrm{中号}}_{ñ}）$ 是同构的 $（ñ，ñ+\mathrm{1个}）$-torus结组–同构 ${\mathcal{乙}}_3$ 为了 $ñ=\mathrm{2个}$ 以及特殊复杂反射群的辫子群 $G_{12}$ 为了 $ñ=3$。这样就产生了一个新的Garside结构$（ñ，ñ+\mathrm{1个}）$-torus结组，已经接受了几种不同的Garside结构。

这 $（ñ，ñ+\mathrm{1个}）$-torus结组是 ${\mathcal{乙}}_{ñ+\mathrm{1个}}$和Garside monoid ${\mathrm{中号}}_{ñ}$ 投射到亚类恐龙上 ${\mathrm{\Sigma }}_{ñ}$ 的 ${\mathcal{乙}}_{ñ+\mathrm{1个}}$ 由...产生 ${\sigma }_{\mathrm{1个}}，{\sigma }_{\mathrm{1个}}{\sigma }_{\mathrm{2个}}，\dots ，{\sigma }_{\mathrm{1个}}{\sigma }_{\mathrm{2个}}\cdots {\sigma }_{ñ}$，当不是Garside monoid时$ñ>\mathrm{2个}$。使用新的演示文稿${\mathcal{乙}}_{ñ+\mathrm{1个}}$ 这类似于 $G（{\mathrm{中号}}_{ñ}）$，不过我们还是检查一下 ${\mathrm{\Sigma }}_{ñ}$ 是矿石单面体，具有成组同构的分数 ${\mathcal{乙}}_{ñ+\mathrm{1个}}$，并给出一个推测性表示，类似于的定义表示 ${\mathrm{中号}}_{ñ}$。这部分回答了Dehornoy–Digne–Godelle–Krammer–Michel的问题。