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 scaleNe, defining comprehensively the macroscopic
rheological properties of the highly entangled polymer systems. Even
though they 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 with various bending stiffness values. Specifically, by introducing
an algorithm that shrinks the chains to their minimal shapes that
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 intrachain level (knots) and
of links between pairs and triplets of distinct chains. Then, by employing
the Z1 algorithm on the minimal conformations to extract the entanglement
lengthNe, we show that the ratioN/Ne, the number of entanglements
per chain, can be remarkably well reconstructed in terms of only two-chain
links.