  Motivated by recent observations that non-concatenated ring polymers in their
dense solution exhibit a glass-like dynamics, we propose a free volume
description of the motion of such rings based on the notion of topological
volume. We first construct a phenomenological free energy which enables one to
qnaitify the degree of topological crowding measured by the coordination
number. Then we pinpoint a key role of the cooperative dynamics of neighboring
rings, which is responsible for an anomalous dependence of the global
structural relaxation (diffusion) time on ring length. Predictions on molecular
weight dependence of both static (ring size, coordination number) and dynamic
(relaxation time, diffusion coefficient) quantities are in very good agreement
with reported numerical simulations. Throughout the discussion, the
entanglement length $N_e$ is assumed to be a unique characteristic length for
the topological constraint, hence, all the physical quantities are universally
described in terms of the rescaled chain length $N/N_e$. Finally, we discuss
how the dense solution of rings is analogous yet different from ordinary glassy
systems.
