Micah Chrisman
The Ohio State University
Title
Thompson's group V and virtual link theory
Abstract
The groups $F \subset T \subset V$ were first introduced by R. Thompson in 1965, and have since had widespread application in fields as diverse as logic, group theory, homotopy theory, link theory, and lattice gauge theory. In 2014, V. F. R. Jones constructed unitary representations of Thompson's groups $F$ and $T$ by factoring through a surjective map from $F$ to isotopy classes of links in the $3$-sphere. Liles extended this to a map from $T$ to the set of checkerboard colorable (CC) links in the thickened annulus. In this talk, we will complete this program for Thompson's group $V$. We show that there is a surjective map $\mathscr{L}_{V}$ from $V$ to the set of CC links in thickened compact oriented surfaces $\Sigma \times [0,1]$, up to L. H. Kauffman's virtual link equivalence relation. The surjection gives rise to a new oriented subgroup $\vec{V} \subset V$, which contains Jones' oriented subgroups $\vec{F} \subset F$ and $\vec{T}\subset T$. We prove that the image $\mathscr{L}_V(\vec{V})$ is the set of oriented virtual links which are almost classical (in the sense of Silver-Williams). We then construct unitary representations of $V$ and $\vec{V}$ from kei coloring invariants and operator quandle coloring invariants, respectively. This generalizes prior work of Aiello-Conti-Jones. The talk discusses joint work with Louisa Liles and Melody Molander.
