Statistical physics of ring polymers based on topological volume concept

Physical properties of ring polymers are subjected to topological constraint, understanding of which has been a significant challenge as a fundamental problem in polymer science. Here we overview a recent attempt to describe behavior of non-concatenated rings in their dense solution based on the notion of topological volume. Discussion includes the size scaling and dynamics of rings, and the phase behavior of blends containing ring polymers. We also discuss the bond correlation function of a ring in two-dimension, for which the non-crossing topological constraint plays a central role.