  The viscous flow of polymer chains in dense melts is dominated by topological
constraints whenever the single chain contour length, N, becomes larger than
the characteristic scale Ne, defining comprehensively the macroscopic
rheological properties of the highly entangled polymer systems. Even though the
latter are naturally connected to the presence of hard constraints like knots
and links within the polymer chains, the difficulty of integrating the rigorous
language of mathematical topology with the physics of polymer melts has limited
somehow a genuine topological approach to the problem of classifying these
constraints and to how they are related to the rheological entanglements. In
this work, we tackle this problem by studying the occurrence of knots and links
in lattice melts of randomly knotted and randomly concatenated ring polymers of
various bending stiffness. Specifically, by introducing an algorithm which
shrinks the chains to their minimal shapes which do not violate topological
constraints and by analyzing those in terms of suitable topological invariants,
we provide a detailed characterization of the topological properties at the
intra-chain level (knots) and of links between pairs and triplets of distinct
chains. Then, by employing the Z1-algorithm on the minimal conformations in
order to extract the entanglement length $N_e$, we show that the ratio $N/N_e$,
the number of entanglements per chain, can be remarkably well reconstructed in
terms of 2-chain links solely.
