On the factorization of composite polymer knots into separated prime components

Using Monte Carlo simulations and advanced knot localization methods, we analyze the length and distribution of prime components in composite knots tied on Freely Jointed Rings (FJRs). For increasing contour length, we observe the progressive factorization of composite knots into separated prime components. However, we observe that a complete factorization, equivalent to the "decorated ring" picture, is not obtained even for rings of contour lengths $N\simeq 3N_0$, about tens of times the most probable length of the prime knots tied on the rings. The decorated ring hypothesis has been used in literature to justify the factorization of composite knots probabilities into the knotting probabilities of their prime components. Following our results, we suggest that such hypothesis may not be necessary to explain the factorization of the knotting probabilities, at least when polymers excluded volume is not relevant. We rationalize the behavior of the system through a simple one dimensional model in which prime knots are replaced by sliplinks randomly placed on a circle, with the only constraint that the length of the loops has the same distribution of the length of the corresponding prime knots.