Non-equilibrium dynamics in untying knots

Flexible closed knots with strict topological constraint of no segment crossing are studied in our simulations using the bond-fluctuation model for polymers. A well-equilibrated polymer knot is cut at a randomly chosen segment and allow to relax to it new equilibrium state, which is the indistinguishable from the free linear chain. In the course of the relaxation, the topological information of the original knot is lost and the characteristic relaxation time measures its rate. The average relaxation time is the typical time scale needed to untie the cut knot by random Brownian motion. Some knot is found to have a longer relaxation time than other knots having more crossings. Remarkably, when all the knots are arranged into homologous groups with a common parametrization in terms of their Alexander polynomials, the relaxation times increase monotonically and linearly with C for all the four groups we studied. Our observation indicates that conventional labelling of knots can further be parametrized naturally into groups in a way that has a direct physical meaning in terms of the topological interactions in a knot. The linear stepwise increase of the relaxation time with the essential crossing suggests that the topological energy spectrum has equal spacing of for knots within a group.