Asymptotics of knotted lattice polygons

We use Monte Carlo methods to investigate the asymptotic behaviour of the number and mean-square radius of gyration of polygons in the simple cubic lattice with fixed knot type. Let p(n)(tau) be the number of n-edge polygons of a fixed knot type tau in the cubic lattice, and let [R-n(2)(tau)] be the mean square radius of gyration of all the polygons counted by p(n)(tau). If we assume that p(n)(tau) similar to n(alpha(tau)-3) mu(tau)(n), where mu(tau) is the growth constant of polygons of knot type tau, and alpha(tau) is the entropic exponent of polygons of knot type tau, then our numerical data are consistent with the relation alpha(tau) = alpha(phi) + N-f, where phi is the unknot and N-f is the number of prime factors of the knot tau. If we assume that [R-n(2)(tau)] similar to A(nu)(tau)n(2 nu(tau)), then our data are consistent with both A(nu)(tau) and nu(tau) being independent of tau. These results support the claims made in Janse van Rensburg and Whittington (1991a J. Phys. A: Math. Gen. 24 3935) and Orlandini er al (1996 J. Phys. A: Math. Gen. 29 L299, 1998 Topology and Geometry in Polymer Science (IMA Volumes in Mathematics and its Applications) (Berlin: Springer)).