论文标题
在圆环组和辫子组的下monoid上
On torus knot groups and a submonoid of the braid group
论文作者
论文摘要
$ 3 $ -Strand Braid $ \ MATHCAL {B} _3 $由$σ_1$和$σ_1σ_2$生成的$ 3 $ -STRAND BRAID GROUP的submonoid已知可以在$ \ Mathcal {B} _3 $上产生异国情调的Garside结构。我们介绍并研究了一个无限的家庭$（m_n）_ {n \ geq 1} $ garside monoids概括了这种异国情调的Garside结构，即$ M_2 $对上述单体是同构的。相应的Garside Group $ g（m_n）$与$（n，n+1）$ - torus结组 - $ \ nathcal {b} _3 $ for $ n = 2 $以及对$ n = 2 $的同构，对$ n = 2 $，并且是异常复杂的复杂反射组$ g_ {12} $ for $ n = 3 $。这产生了$（n，n+1）$ - 圆环组的新的Garside结构，该结构已经接收了几个不同的Garside结构。 $（n，n+1）$ - 圆环组是$ \ Mathcal {b} _ {n+1} $的延伸，而Garside monoid $ m_n $向下$ \ \ \ \ \ \ \ natercal {b} _ {b} _ {n+1} $ $ n+σ__1的$ \ n $σ_1$σ__1 σ_2\cdotsσ_n$，当$ n> 2 $时，它不是garside monoid。使用$ \ Mathcal {b} _ {n+1} $的新演示文稿类似于$ g（m_n）$的呈现，我们仍然检查$σ_n$是一个矿石单型矿石，带有一组与$ \ $ \ \ \ \ \ \ \ mathcal {b} _ {n+1} $的分数，并给出了一个interations contresition in Indistion in Itderations，并给出了一个interation in Iteforation in Iteforation in Iteforation an $ m_n $。这部分回答了DeHornoy-Digne-Godelle-Krammer-Michel的问题。
The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $σ_1$ and $σ_1 σ_2$ 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 $Σ_n$ of $\mathcal{B}_{n+1}$ generated by $σ_1, σ_1 σ_2, \dots, σ_1 σ_2\cdots σ_n$, 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 $Σ_n$ 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.
