Knotting statistics for polygons in lattice tubes

Abstract

We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite transfer matrix, and this allows for the derivation of pattern theorems, calculation of growth rates and exact enumeration. We also develop a static Monte Carlo method which allows us to sample polygons of a given size directly from a chosen Boltzmann distribution. Using these methods we accurately estimate the growth rates of unknotted polygons in the 2 × 1×∞ and 3 × 1×∞ tubes, and confirm that these are the same for any fixed knot-type K. We also confirm that..