T. Deguchi et K. Tsurusaki, UNIVERSALITY OF RANDOM KNOTTING, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 55(5), 1997, pp. 6245-6248
Knotting probability [P-K(N)] is defined by the probability of an N-no
ded random polygon being topologically equivalent to a given knot K. F
or several nontrivial knots we numerically evaluate the knotting proba
bilities for Gaussian and rod-bead models. We find that they are well
approximated by the following formula: P-K(N)=C(K)[(N) over tilde/N(K)
](m(K))exp[-(N) over tilde/N(K)] where (N) over tilde=N-N-ini(K), and
that the fitting parameters C(K); N(K), and N-ini(K) are model depende
nt, 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.