Virtual Thompson's group

TL;DR

The paper introduces the virtual Thompson's group $VF$ as a diagram group over a finite semigroup, generalizing Thompson's group $F$ to virtual knot theory. It proves an Alexander-type theorem: every virtual link can be constructed from an element of $VF$ using a three-step construction from diagrams to Thompson graphs to medial graphs, yielding $L(\Delta)$ for some $\Delta\in VF$. The work connects $VF$ to labeled binary trees and Gamma-graphs, extending Jones's $F$-construction to $VF$ and providing an infinite presentation via the Squier complex, while noting potential oriented extensions with $\overrightarrow{VF}$. These results bridge virtual knot theory with diagram-group techniques, highlighting algebraic and geometric properties inherited from diagram groups and opening avenues for oriented and higher-structure generalizations.

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.

Figures (21)

• Figure 1: Classical and virtual crossings
• Figure 4: Classical Reidemeister moves
• Figure 8: Virtual Reidemeister moves
• Figure 12: Mixed move
• Figure 13: Trivial geometric diagrams of $a$ and $ab$, and geometric $(a, ab)$-cell and $(aa, a)$-cell