We review recent theoretical studies on topological properties of ring polymers in solution. The topology of a ring polymer corresponds to a knot type and is conserved during time evolution. As a typical topological effect we consider topological swelling: The mean square radius of gyration of a ring polymer in solution becomes larger than that of an ideal phantom ring polymer, which has no topological constraint, at the theta temperature of the corresponding linear polymer. Here, at the theta temperature the excluded volume of the linear polymer vanishes effectively. The expansion of the average size is a consequence of the entropic repulsive force due to the topological constraint. We show that topological swelling occurs only for ring polymers with small excluded volume. Furthermore, several topological effects predicted theoretically can now be confirmed directly by experiments in terms of synthetic ring polymers.
