Universality of random knotting

Knotting probability [${\mathrm{P}}_{\mathrm{K}}$(N)] is defined by the probability of an N-noded random polygon being topologically equivalent to a given knot K. For several nontrivial knots we numerically evaluate the knotting probabilities for Gaussian and rod-bead models. We find that they are well approximated by the following formula: ${\mathrm{P}}_{\mathrm{K}}$(N)=C(K)[Ñ/N(K)${]}^{\mathrm{m}\left(\mathrm{K}\right)\mathrm{}}$exp[-Ñ/N(K)] where Ñ=N-${\mathrm{N}}_{\mathrm{ini}}$(K), and that the fitting parameters C(K), N(K), and ${\mathrm{N}}_{\mathrm{ini}}$(K) are model dependent, while m(K) is not. We suggest that given a knot K, the exponent m(K) should be universal: it is independent of models of random polygon and is determined only by the knot K.