Monte Carlo simulations are performed to study the equilibrium structure and nonequilibrium dynamic relaxation processes of knotted polymers. We find that topological complexity affects the static and dynamic behavior of knots in different ways to different extent. For statics, our results on the radii of gyration of knot polymers suggest that prime and two-factor composite knots belong to different groups, and we confirm that for knots in the same group, the average radius of gyration scales as Rg∼N3/5p−4/15 in good solvents, where N is the number of monomers and p is the topological invariant representing the length-to-diameter ratio of the knot at its maximum inflated state. From the studies of nonequilibrium relaxation dynamics on prime knots cut at t=0, we find that even prime knots should be classified into different groups as (31,51,71,…),(41,61,81,…),(52,72,92,…), etc., based on their topological similarity and their polynomial invariants such as Alexander polynomials. Our results suggest that the mathematical classification of knots can further be parametrized naturally into groups in a way that can have direct physical meaning in terms of structures and dynamics of knots. Furthermore, by scaling calculations, the nonequilibrium relaxation time is found to increase roughly as p12/5. This prediction is further supported by our data.
