We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

中文翻译：

---

关于一些虚结群的下中心系列

我们研究了一些具有少量交叉的虚拟结的群，并证明了存在一个具有较长下中心系列的虚拟结，这尤其意味着存在一个具有剩余幂零群的虚拟结。这提供了一种可能性，可以使用根据结组的下中心系列的商来构造虚拟结的不变量。此外，我们研究了虚拟结组分解为半直积和自由积的合并。特别是，我们证明了一些虚拟结的群是由无限循环群对有限生成的自由群的扩展。