Size of knots in ring polymers

We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modeled by a self avoiding polygon of N steps on cubic lattice and l is the number of steps over which the knot "spreads" in a given configuration. An analysis of extensive Monte Carlo data in equilibrium shows that the probability distribution of l as a function of N obeys a scaling of the form p(l,N)similar to l(-c)f(l/N(D)), with c similar or equal to 1.25 and D similar or equal to 1. Both D and c could be independent of knot type. As a consequence, the knot is weakly localized, i.e., < l >similar to N(t), with t=2-c similar or equal to 0.75. For a ring with fixed knot type, weak localization implies the existence of a peculiar characteristic length l(nu)similar to N(t nu). In the scaling similar to N(nu) (nu similar or equal to 0.58) of the radius of gyration of the whole ring, this length determines a leading power law correction which is much stronger than that found in the case of unrestricted topology. The existence of this correction is confirmed by an analysis of extensive Monte Carlo data for the radius of gyration. The collapsed regime is studied by introducing in the model sufficiently strong attractive interactions for nearest neighbor sites visited by the self-avoiding polygon. In this regime knot length determinations can be based on the entropic competition between two knotted loops separated by a slip link. These measurements enable us to conclude that each knot is delocalized (t similar or equal to 1).