  Understanding how topological constraints affect the dynamics of polymers in
solution is at the basis of any polymer theory and it is particularly needed
for melts of rings. These polymers fold as crumpled and space-filling objects
and, yet, they display a large number of topological constraints. To understand
their role, here we systematically probe the response of solutions of rings at
various densities to "random pinning" perturbations. We show that these
perturbations trigger non-Gaussian and heterogeneous dynamics, eventually
leading to non-ergodic and glassy behaviours. We then derive universal scaling
relations for the values of solution density and polymer length marking the
onset of vitrification in unperturbed solutions. Finally, we directly connect
the heterogeneous dynamics of the rings with their spatial organisation and
mutual interpenetration. Our results suggest that deviations from the typical
behaviours observed in systems of linear polymers may originate from
architecture-specific (threading) topological constraints.
