Geometry and topology of knotted ring polymers in an array of obstacles

We study knotted polymers in equilibrium with an array of obstacles which models confinement in a gel or immersion in a melt. We find a crossover in both the geometrical and the topological behavior of the polymer. When the polymers' radius of gyration, R(G), and that of the region containing the knot, R(G,k), are small compared to the distance b between the obstacles, the knot is weakly localized and R(G) scales as in a good solvent with an amplitude that depends on knot type. In an intermediate regime where R(G) > b > R(G,k), the geometry of the polymer becomes branched. When R(G,k) exceeds b, the knot delocalizes and becomes also branched. In this regime, R(G) is independent of knot type. We discuss the implications of this behavior for gel electrophoresis experiments on knotted DNA in weak fields.