  We present a computer simulation study of the compact self-avoiding loops as
regards their length and topological state. We use a Hamiltonian closed path on
the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer.
The importance of ergodic sampling of all loops is emphasized. We first look at
the effect of global topological constraint on the local fractal geometry of a
typical loop. We find that even short pieces of a compact trivial knot, or some
other under-knotted loop, are somewhat crumpled compared to topology-blind
average over all loops. We further attempt to examine whether knots are
localized or de-localized along the chain when chain is compact. For this, we
perform computational decimation and chain coarsening, and look at the
"renormalization trajectories" in the space of knots frequencies. Although not
completely conclusive, our results are not inconsistent with the idea that
knots become de-localized when the polymer is compact.
