On the universality of knot probability ratios

Let pn denote the number of self-avoiding polygons of length n on a regular three-dimensional lattice, and let pn(K) be the number which have knot type K. The probability that a random polygon of length n has knot type K is pn(K)/pn and is known to decay exponentially with length [1, 2]. Little is known rigorously about the asymptotics of pn(K), but there is substantial numerical evidence [3, 4, 5, 6] that pn(K) grows as pn(K) ' CK μ∅ n α−3+NK , as n→∞, where NK is the number of prime components of the knot type K. It is believed that the entropic exponent, α, is universal, while the exponential growth rate, μ∅, is independent of the knot type but varies with the lattice. The amplitude, CK , depends on both the lattice and the knot type. The above asymptotic form implies that the relative probability of a random polygon of length n having prime knot type K over prime knot type L is pn(K)/pn pn(L)/pn = pn(K) pn(L) ' [ CK CL ]