  A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does
not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N.
Thurston identified seven families of exceptional manifolds in their proof of
the homotopy rigidity theorem. They identified the hyperbolic manifold known as
Vol3 in the literature as the exceptional manifold associated to one of the
families. It is conjectured that there are exactly 6 exceptional manifolds.
  We find hyperbolic 3-manifolds, some from the SnapPea's census of closed
hyperbolic 3-manifolds, associated to 5 other families. Along with the
hyperbolic 3-manifold found by Lipyanskiy associated to the seventh family we
show that any exceptional manifold is covered by one of these manifolds. We
also find their group coefficient fields and invariant trace fields.
