The knot probability in lattice polygons

Abstract

The incidence of knots in lattice polygons in the face-centred cubic lattice is investigated numerically. The authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the Alexander polynomial. If p0n( phi ) is the probability that the polygon with n edges is the unknot, then it is known that lim supn to infinity p0n( phi )1n/=e- varies as (0)<1. They find that varies as ( phi )=(7.6+or-0.9)*10-6. The effect of the solvent quality on p0n( phi ) is considered. The data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates.