  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 modelled 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) ~ l^(-c) f(l/N^D), with c ~ 1.25 and D ~ 1. Both D and c could
be independent of knot type. As a consequence, the knot is weakly localized,
i.e. <l> ~ N^t, with t=2-c ~ 0.75. For a ring with fixed knot type, weak
localization implies the existence of a peculiar characteristic length l^(nu) ~
N^(t nu). In the scaling ~ N^(nu) (nu ~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
such 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 ~ 1).
