New evidence on the asymptotics of knotted lattice polygons via local strand-passage models

Abstract

Two simple-cubic-lattice models of local strand passage in self-avoiding polygons are used to study the asymptotic properties of knot probabilities. Using Monte Carlo simulations of fixed knot-type, variable-length polygons for knot-types , we investigate the limiting (as polygon length goes to ∞) knot-transition probability to go from knot-type K to knot-type K′ after a single strand passage. For knot-types K and Kb, we present numerical evidence that the model-dependent limiting knot-transition probability to go from knot-type K to K′ = K#Kb is independent of the original knot-type K, and that the limiting knot-transition probability is zero for any transition from K to a knot K′ which does not have K in its prime-knot decomposition. These results provide new numerical evidence in support of the conjectured asymptotic form, , for the number of n-edge polygons with fixed knot-type K, where α0 and μ0 are respectively the entropic critical exponent and limiting entropy per edge for unknotted lattice polygons and fK is the number of prime knots in K’s prime-knot decomposition.