论文标题
在虚拟单数编织组上
On virtual singular braid groups
论文作者
论文摘要
令$ n \ ge 2 $。在本文中，我们研究了虚拟单数编织组$ vsg_n $的一些属性。我们为虚拟单数辫子定义了一些数字不变性，这些辫子是由$ vsg_n $中的单词指数引起的，并描述了这些同构的核。我们从虚拟单数编织组$ vsg_n $到对称组$ s_n $确定所有可能的同态，直到共轭，而对于特定情况$ n = 2 $，我们在每种情况下为内核提供了演示文稿（和描述）。对于所有可能的同态，我们获得了$ vsg_n $的分解，作为同构和对称组的核的半导体产品。最后，我们表明$ vsg_n $中有一些禁止的关系，然后我们介绍了一些商（例如，焊接的奇异辫子组，不受限制的虚拟单数辫子组，以及其他相关小组），然后我们对它们表示相似的结果，以$ vsg_n $获得。
Let $n\ge 2$. In this paper we study some properties of the virtual singular braid group $VSG_n$. We define some numerical invariants for virtual singular braids arising from exponents of words in $VSG_n$ and describe the kernel of these homomorphisms. We determine all possible homomorphisms, up to conjugation, from the virtual singular braid group $VSG_n$ to the symmetric group $S_n$, and for the particular case $n=2$ we give a presentation (and a description) for the kernel in each case. For all possible homomorphisms we obtain a decomposition of $VSG_n$ as a semi-direct product of the kernel of the homomorphism and the symmetric group. Finally, we show that there are some forbidden relations in $VSG_n$ and then we introduce some quotients of it (e.g. the welded singular braid group, the unrestricted virtual singular braid group, among other related groups) and then we state for them similar results obtained for $VSG_n$.
