On Jones Subgroup of R. Thompson's Group $T$

Jones introduced unitary representations for the Thompson groups $F$ and $T$ from a given subfactor planar algebra. Some interesting subgroups arise as the stabilizer of certain vector, in particular the Jones subgroups $\vec{F}$ and $\vec{T}$. Golan and Sapir studied $\vec{F}$ and identified it as a copy of the Thompson group $F_3$. In this paper we completely describe $\vec{T}$ and show that $\vec{T}$ coincides with its commensurator in $T$, implying that the corresponding unitary representation is irreducible. We also generalize the notion of the Stallings 2-core for diagram groups to $T$, showing that $\vec{T}$ and $T_3$ are not isomorphic, but as annular diagram groups they have very similar presentations.