Thompson's Group $V$ and Virtual Link Theory

Abstract

Thompson's groups $F \subset T \subset V$ were introduced in 1965 and have since found widespread application in fields as diverse as logic, group theory, homotopy theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of $F$, factoring through a surjection from $F$ to isotopy classes of links in $S^3$. The second author extended Jones' surjection to $T$, thereby constructing all isotopy classes of checkerboard colorable (CC) links in the thickened annulus. We complete this program for $V$, defining a surjection $\mathcal{L}_{V}$ from $V$ to virtual equivalence classes of CC links in thickened compact oriented surfaces. This yields a new oriented subgroup $\vec{V} \subset V$ containing Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove $\vec{V}$ realizes all oriented almost classical virtual links. We then construct unitary representations of $V$ and $\vec{V}$ from kei and operator quandle coloring invariants, respectively.