We employ extensive computer simulations to investigate the conformations and the interactions of ring polymers under conditions of worsening solvent quality, in comparison with those for linear polymers. We determine the dependence of the Theta-temperature on knotedness by considering ring polymers of different topologies. We establish a clear decrease of the former upon changing the topology of the polymer from linear to an unknotted ring and a further decrease of the same upon introducing trefoil- or 5-fold knots but we find no difference in the Theta-point between the two knotted molecules. Our results are based on two independent methods: one considering the scaling of the gyration radius with molecular weight and one based on the dependence of the effective interaction on solvent quality. In addition, we calculate several shape-parameters of the polymers to characterize linear, unknotted, and knotted topologies in good solvents and in the proximity of the Theta-point. The shape parameters of the knotted molecules show an interesting crossover at a degree of polymerization that depends on the degree of knottedness of the molecule.