Abstract
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 be the number of n-edge polygons of a fixed knot type in the cubic lattice, and let be the mean square radius of gyration of all the polygons counted by . If we assume that , where is the growth constant of polygons of knot type , and is the entropic exponent of polygons of knot type , then our numerical data are consistent with the relation , where is the unknot and is the number of prime factors of the knot . If we assume that , then our data are consistent with both and being independent of . These results support the claims made in Janse van Rensburg and Whittington (1991a 24 3935) and Orlandini et al (1996 29 L299, 1998 Topology and Geometry in Polymer Science (IMA Volumes in Mathematics and its Applications) (Berlin: Springer)).
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