Non-monotonicity in the knotting probability of semiflexible rings: numerical and analytical prediction

Polymers in canonical equilibrium are prone to become knotted, and the knotting probability for a specific type of knot depends on several conditions, among these we found interesting to study on the effect of bending rigidity of self-avoiding polymers. We use a simple physical mapping to adapt the known asymptotic expressions for the knotting probabilities of self-avoiding polygons to the case of semiflexible rings of beads. We thus obtain analytical expressions that approximate the abundance of the simplest knots as a function of the length and bending rigidity of the rings. We validate the predictions against previously published data from stochastic simulations of rings of beads showing that they reproduce the intriguing non-monotonic dependence of knotting probability on bending rigidity. The mapping thus provides a useful theoretical tool not only for a physically-transparent interpretation of previous results, but especially to predict the rigidity-dependent knotting probabilities for previously unexplored combinations of chain lengths and bending rigidities. In particular, our mapping suggests that for rings longer than 20,000 beads, the rigidity-dependent knotting probability prole switches from unimodal to bimodal.