  Pure reconstruction phases, geometric and dynamic, are computed in the
$N$-point-vortex model in the plane, for the cases $N=3$ and $N=4$. The phases
are computed relative to a metric-orthogonal connection on appropriately
defined principal fiber bundles. The metric is similar to the kinetic energy
metric for point masses but with the masses replaced by vortex strengths. The
geometric phases are shown to be proportional to areas enclosed by the closed
orbit on the symmetry reduced spaces. More interestingly, simple formulae are
obtained for the dynamic phases, analogous to Montgomery's result for the free
rigid body, which show them to be proportional to the time period of the
symmetry reduced closed orbits. For the case $ N = 3 $ a non-zero total vortex
strength is assumed. For the case $ N = 4 $ the vortex strengths are assumed
equal.
