Topological effects on statics and dynamics of knotted polymers

Using dynamic Monte Carlo simulations, our results on the radii of gyration of knot polymers suggest that prime and two-factor composite knots belong to different groups. From the studies of nonequilibrium relaxation dynamics on cut prime knots, we find that even prime knots should be classified into different groups, such as ${(3\mathrm{}}_1{,5\mathrm{}}_1,\dots ),$ ${(4\mathrm{}}_1{,6\mathrm{}}_1,\dots ),$ and ${(5\mathrm{}}_2{,7\mathrm{}}_2,\dots ),$ etc., based on their topological similarity and their polynomial invariants. By scaling calculations, the nonequilibrium relaxation time is found to increase roughly as $p^{12/5},$ where $p$ is the topological invariant length-to-diameter ratio of the knot at its maximum inflated state. This prediction is further confirmed by our data.