On the universality of knot probability ratios

Abstract

Let $p_n$ denote the number of self-avoiding polygons of length $n$ on a regular three-dimensional lattice, and let $p_n(K)$ be the number which have knot type $K$. The probability that a random polygon of length $n$ has knot type $K$ is $p_n(K)/p_n$ and is known to decay exponentially with length. Little is known rigorously about the asymptotics of $p_n(K)$, but there is substantial numerical evidence that $p_n(K)$ grows as $p_n(K) \simeq \, C_K \, \mu_\emptyset^n \, n^{\alpha-3+N_K}$, as $n \to \infty$, where $N_K$ is the number of prime components of the knot type $K$. It is believed that the entropic exponent, $\alpha$, is universal, while the exponential growth rate, $\mu_\emptyset$, is independent of the knot type but varies with the lattice. The amplitude, $C_K$, 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 $\frac{p_n(K)/p_n}{p_n(L)/p_n} = \frac{p_n(K)}{p_n(L)} \simeq [ \frac{C_K}{C_L} ]$. In the thermodynamic limit this probability ratio becomes an amplitude ratio; it should be universal and depend only on the knot types $K$ and $L$. In this letter we examine the universality of these probability ratios for polygons in the simple cubic, face-centered cubic, and body-centered cubic lattices. Our results support the hypothesis that these are universal quantities. For example, we estimate that a long random polygon is approximately 28 times more likely to be a trefoil than be a figure-eight, independent of the underlying lattice, giving an estimate of the intrinsic entropy associated with knot types in closed curves.