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We study knotted polymers in equilibrium with an array of obstacles whichmodels confinement in a gel or immersion in a melt. We find a crossover in boththe 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 knotis weakly localised and $R_G$ scales as in a good solvent with an amplitudethat 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, theknot delocalises and becomes also branched. In this regime, $R_G$ isindependent of knot type. We discuss the implications of this behavior for gelelectrophoresis experiments on knotted DNA in weak fields.

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