Gaussian statistics of random walks and/or "ideal" polymers is
heavily dependent on the presence of configurations with knots.
If knots are excluded by, e.g., loop preparation, then such loop
is not Gaussian.  Extensive simulation confirms earlier
predictions that no-knots loop has the statistics similar to that
of a self-avoiding walk, even if real excluded volume is
completely negligible.  Thus, global topological constraint leads
to local polymer swelling for a free loop.  By contrast, compact
polymer loop constrained to have no knots and to occupy minimal
possible volume exhibits opposite effect, with every piece of a
polymer somewhat shrunken.