Effects of Knotting on the Collapse of Active Ring Polymers

We use numerical simulations to study tangentially active flexible ring polymers with different knot topologies. Simple, unknotted active rings display a transition from an extended phase to a collapsed one upon increasing the degree of polymerization. We find that topology has a significant effect on the polymer size at which the collapse takes place, with twist knots collapsing earlier than torus knots. Increasing knot complexity further accentuates this difference, as the collapse point of torus knots grows linearly with the minimum crossing number of the knot while that of twist knots shrinks, eventually canceling the actively stretched regime altogether. This behavior is a consequence of the ordered configuration of torus knots in their stretched active state, featuring an effective alignment for non-neighboring bonds which increases with the minimal crossing number. Twist knots do not feature ordered configurations or bond alignment, increasing the likelihood of collisions, leading to collapse. These results show that topology yields a degree of control on the properties of active ring polymers, and can be used to tune them. At the same time, they suggest that activity might introduce a bias for torus knots, as complex twist knots cannot be formed in extended active polymers.