Bimodality in the knotting probability of semiflexible rings suggested by mapping with self-avoiding polygons

We use a simple physical mapping to adapt the known asymptotic expressions for the knotting probabilities ofself-avoiding polygons to the case of semiflexible rings of beads. We thus obtain analytical expressions thatapproximate the abundance of the simplest knots as a function of the length and bending rigidity of the rings. Wevalidate the predictions against previously published data from stochastic simulations of rings of beads showingthat they reproduce the intriguing non-monotonic dependence of knotting probability on bending rigidity. Themapping thus provides a useful theoretical tool not only for a physically-transparent interpretation of previousresults, but especially to predict the knotting probabilities for previously unexplored combinations of chainlengths and bending rigidities. In particular, our mapping suggests that for rings longer than 20,000 beads, therigidity-dependent knotting probability profile switches from unimodal to bimodal