Knottedness in ring polymers

We represent ring polymers in free space with the rod-bead model and show through unbiased computer simulations that the probability of observing a trivial self-entanglement (P) has a decreasing exponential dependence on the contour length (N) of the polymer, or that P=exp(-N/${\mathit{N}}_0$). The characteristic length (${\mathit{N}}_0$) varies by many orders of magnitude depending on chain flexibility and solvent quality. We also suggest that sufficiently large knots are always composite, not prime.