Topology-dependent mixing of knots in flexible polymer chains
The organization of multiple knots along a polymer contour can depend sensitively on the topology of the constituent knots. Using coarse-grained Brownian-dynamics simulations, we investigate topology-dependent mixing of two knots in a flexible polymer at controlled normalized wall separation. We compare homogeneous ($3_1\otimes 3_1$) and heterogeneous ($3_1\otimes 5_1$) knot pairs using a contour-based coordinate that distinguishes mixed and demixed configurations. Starting from partially overlapping configurations, the $3_1\otimes 5_1$ pair is observed in mixed configurations more frequently over the simulation time window than the $3_1\otimes 3_1$ pair at intermediate wall separation. We further find that, over the intermediate range of normalized wall separation, mixing of the $3_1\otimes5_1$ pair produces a composite knotted region that occupies substantially less polymer contour than the two constituent knots collectively occupy in demixed configurations. The corresponding reduction in contour occupancy is much smaller for the $3_1\otimes3_1$ pair. This indicates that the heterogeneous constituent topologies can be accommodated within a smaller portion of the polymer contour in the composite configuration. These results establish that constituent topology can strongly influence multiple-knot organization in flexible polymers.