The dimensions of knotted polygons

Abstract

The authors study the dimensions (mean-square radius of gyration and mean span) of self-avoiding polygons on the simple cubic lattice with fixed knot type. The approach used is a Monte Carlo algorithm which is a combination of the BFACF algorithm and the pivot algorithm, so that the polygons are studied in the grand canonical ensemble, but the autocorrelation time is not too large. They show that, although the dimensions of polygons are sensitive to knot type, the critical exponent (v) and the leading amplitude are independent of the knot type of the polygon. The knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour.