  We perform large scale three-dimensional molecular dynamics simulations of
unlinked and unknotted ring polymers diffusing through a background gel, here a
three-dimensional cubic lattice. Taking advantage of this architecture, we
propose a new method to unambiguously identify and quantify inter-ring
threadings (penetrations) and to relate these to the dynamics of the ring
polymers. We find that both the number and the persistence time of the
threadings increase with the length of the chains, ultimately leading to a
percolating network of inter-ring penetrations. We discuss the implications of
these findings for the possible emergence of a topological jammed state of very
long rings.
