Manifolds with the same quantum invariants

There exist infinitely many pairs of non-homeomorphic SOL manifolds with the same
quantum invariants. This follows from the arithmetic behind the conjugacy problem in SL(2,Z) and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementary study of a family of Pell equations. On the other hand two torus bundles over the circle with the same quantum invariants are (strongly) commensurable.