Virtual Thompson's group

TL;DR

This paper introduces a virtual version of Thompson's group F and proves that any virtual link can be represented using elements of this group, extending classical link construction methods.

Contribution

It defines the virtual Thompson's group and establishes its capability to generate all virtual links, bridging virtual knot theory with group theory.

Findings

Defined the virtual Thompson's group F_v

Proved all virtual links can be constructed from F_v elements

Extended classical link construction methods to virtual links

Abstract

For virtual knot theory, the virtual braid group was defined by generalizing the braid group. It was proved that any virtual link can be obtained by the closure of a virtual braid. On the other hand, due to work by Jones et al., it is known that any (oriented) link is constructed from an element of Thompson's group $F$. In this paper, we define the ``virtual version'' of Thompson's group $F$ and prove that any virtual link is constructed from an element of the group.