Virtual Thompson's group

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.