On the universality of knot probability ratios

Abstract. 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 µn ∅ nα−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