Groups of virtual trefoil and Kishino knots

TL;DR

This paper investigates specific groups associated with virtual trefoil and Kishino knots, revealing their structures, non-isomorphisms, and potential for constructing invariants for virtual knots.

Contribution

It analyzes the structure of three groups related to virtual knots, proves non-isomorphism among them, and demonstrates how these groups can be used to create invariants for virtual knots.

Findings

Some groups are not isomorphic to each other

G_3 distinguishes Kishino knot from trivial knot

Groups have non-stabilizing lower central series

Abstract

In the paper of Yu. A. Mikhalchishina for an arbitrary virtual link $L$ three groups $G_{1,r}(L)$, $r>0$, $G_{2}(L)$ and $G_{3}(L)$ were defined. In the present paper these groups for the virtual trefoil are investigated. The structure of these groups are found out and the fact that some of them are not isomorphic to each other is proved. Also we prove that $G_3$ distinguishes the Kishino knot from the trivial knot. The fact that these groups have the lower central series which does not stabilize on the second term is noted. Hence we have a possibility to study these groups using quotients by terms of the lower central series and to construct representations of these groups in rings of formal power series. It allows to construct an invariants for virtual knots.