Nonequilibrium relaxation times in polymer knot groups

Flexible polymer knots with strict topological constraint of no segment crossing are studied by Monte Carlo simulations. The nonequilibrium relaxation time of an equilibrated polymer knot cut at one point to relax to a linear chain is measured. Prime knots up to 20 essential crossings from the groups (3(1),5(1),7(1)....), (4(1), 6(1), 8(1),...) and (5(2), 7(2), 9(2)....), (6(2), 8(2), 10(2)....) are studied. The nonequilibrium relaxation time for knots within a group are found to increase stepwise linearly with the number of essential crossings of the original knot. Our results suggest an equally spaced topological interaction energy spectrum for knots in the same group and thus provide a quantitative description of topological interactions.