Tight and loose shapes in flat entangled dense polymers.

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.