 SU2 character variety of the punctured torus

The punctured torus retracts onto two circles, so its flat $SU(2)$ gauge-orbit space is $SU(2{)}^2/SU(2)$ under simultaneous conjugation. Write unit quaternions as $A=a+u$ and $B=b+v$. Complete orbit invariants are $(a,b,u\cdot v)$ with $(u\cdot v{)}^2\le (1-a^2)(1-b^2)$. Adding $ab$ to the third coordinate identifies this space with the compact convex set of positive-semidefinite $3\times 3$ real correlation matrices, a closed three-ball. The dense open subset $A\ne \pm 1$ is a closed disk times an open interval; its complement is two disjoint intervals.

// Destination: geometry-and-topology.bigb