Under-knotted and over-knotted polymers: compact self-avoiding loops
  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. 
R. C. Lua, N. T. Moore, A. Yu. Grosberg