  In the first part of this work a summary is provided of some recent
experiments and theoretical results which are relevant in the research of
systems of polymer rings in nontrivial topological conformations. Next, some
advances in modeling the behavior of single polymer knots are presented. The
numerical simulations are performed with the help of the Wang-Landau Monte
Carlo algorithm. To sample the polymer conformation a set of random
transformations called pivot moves is used. The crucial problem of preserving
the topology of the knots after each move is tackled with the help of two new
techniques which are briefly explained. As an application, the results of an
investigation of the effects of topology on the thermal properties of polymer
knots is reported. In the end, original results are discussed concerning the
use of parallelized codes to study polymers knots composed by a large number of
segments within the Wang-Landau approach.
