  We investigate the effects of topological constraints (entanglements) on two
dimensional polymer loops in the dense phase, and at the collapse transition
(Theta point). Previous studies have shown that in the dilute phase the
entangled region becomes tight, and is thus localised on a small portion of the
polymer. We find that the entropic force favouring tightness is considerably
weaker in dense polymers. While the simple figure-eight structure, created by a
single crossing in the polymer loop, localises weakly, the trefoil knot and all
other prime knots are loosely spread out over the entire chain. In both the
dense and Theta conditions, the uncontracted knot configuration is the most
likely shape within a scaling analysis. By contrast, a strongly localised
figure-eight is the most likely shape for dilute prime knots. Our findings are
compared to recent simulations.
