  Several nontrivial properties are shown for the mean square radius of
gyration $R_K^2$ of ring polymers with a fixed knot type K.
  Through computer simulation, we discuss both finite-size and asymptotic
behaviors of the gyration radius under the topological constraint for
self-avoiding polygons consisting of N cylindrical segments with radius r. We
find that the average size of ring polymers with a knot K can be much larger
than that of no topological constraint. The effective expansion due to the
topological constraint depends strongly on the parameter r which is related to
the excluded volume. The topological expansion is particularly significant for
the small r case, where the simulation result is associated with that of random
polygons with the knot K.
