  The bond fluctuation method is used to simulate both non-concatenated
entangled and interpenetrating melts of ring polymers. We find that the
swelling of interpenetrating rings upon dilution follows the same laws as for
linear chains. Knotting and linking probabilities of ring polymers in
semi-dilute solution are analyzed using the HOMFLY polynomial. We find an
exponential decay of the knotting probability of rings. The correlation length
of the semi-dilute solution can be used to superimpose knotting data at
different concentrations. A power law dependence $f_{n}\sim\phi
R^{2}\sim\phi^{0.77}N$ for the average number $f_{n}$ of linked rings per ring
at concentrations larger than the overlap volume fraction of rings $\phi^{*}$
is determined from the simulation data. The fraction of non-concatenated rings
displays an exponential decay $P_{OO}\sim\exp(-f_{n})$, which indicates $f_{n}$
to provide the entropic effort for not forming concatenated conformations.
Based upon this results we find four different regimes for the conformations of
rings in melts that are separated by a critical lengths $N_{OO}$, $N_{C}$ and
$N^{*}$. $N_{OO}$ describes the onset of the effect of non-concatenation below
which topological effects are not important, $N_{C}$ is the cross-over between
weak and strong compression of rings, and $N^{*}$ is defined by the cross-over
from a non-concatenation contribution $f_{n}\sim\phi R^{2}$ to an overlap
dominated concatenation contribution $f_{n}\sim\phi N^{1/2}$ at $N>N^{*}$. For
$N_{OO}<N<N_{C}$, the scaling of ring sizes $R\sim N^{2/5}$ results from
balancing non-concatenation with weak compression of rings. For
$N_{C}<N<N^{*}$, non-concatenation and strong compression imply $R\sim
N^{3/8}$. Our simulation data for non-interpenetrating rings up to $N=1024$ are
in good agreement with the prediction for weakly compressed rings.
