Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs "topological polymers". Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology "topological polymers", for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer⁻Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it "topological swelling".