"Understanding the Physics of modern materials on a fundamental level is a key ingredient in the quest to develop novel materials with desired, tailored properties. In this sense, polymers, are in enormous interest nowadays, due to their numerous applications in biology and industry. The development of experimental techniques, such as atomic force microscopy (AFM) and optical tweezers, allow us to manipulate the topology of macromolecules, including the most common biological polymer, DNA. Topological constraints on the polymers are an universal feature that can influence the properties of individual macromolecules and solutions of the same. Their analytical consideration is hindered, i.a., by the difficulty to include topological constraints in the Hamiltonian. Computer simulations offer an alternative. Previous work has focused, mainly, in knotting probabilities with lattice models in three and two dimensional models. We propose the approach of performing computer simulations in the continuum, to investigate the structural properties of ring polymer solutions with topological constraints and predict the correlations, structure and thermodynamics of the system. To accomplish this project, we will develop suitable coarse-grained techniques and advanced programming to sample the properties of ring polymer solutions in ways that respect the topological constraints. Further, we propose a novel development of the blob technique for topological constraints in dilute and semidilute solutions. The goals of the proposal is, on the one hand, to calculate accurate effective interaction potentials acting between suitably chosen coarse-grained degrees of freedom of the rings, greatly facilitating thereby their study, and to study the effects of knots on the quantitative characteristics of the effective potential and thus on the structure and thermodynamics."