Atmospheres of polygons and knotted polygons

Abstract

In this paper we define two statistics a+(ω) and a−(ω), the positive and negative atmospheres of a lattice polygon ω of fixed length n. These statistics have the property that ⟨a+(ω)⟩/⟨a−(ω)⟩ = pn+2/pn, where pn is the number of polygons of length n, counted modulo translations. We use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres. Using these data, we directly estimate the growth constants of polygons in two and three dimensions. We find that where the error bars are 67% confidence intervals, and the second error bar in the three-dimensional estimate of μ is an estimated systematic error. We also compute atmospheres of polygons of fixed knot type K sampled by the BFACF algorithm. We discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n.