  It is nontrivial whether the average size of a ring polymer should become
smaller or larger under a topological constraint.
  Making use of some knot invariants, we evaluate numerically the mean square
radius of gyration for ring polymers having a fixed knot type, where the ring
polymers are given by self-avoiding polygons consisting of freely-jointed hard
cylinders. We obtain plots of the gyration radius versus the number of
polygonal nodes for the trivial, trefoil and figure-eight knots. We discuss
possible asymptotic behaviors of the gyration radius under the topological
constraint. In the asymptotic limit, the size of a ring polymer with a given
knot is larger than that of no topological constraint when the polymer is thin,
and the effective expansion becomes weak when the polymer is thick enough.
