Fig. 1.

Networks of threadings. (A) Snapshot from a simulation representing a system with $N=50$ chains $M=512$ beads long. (B) Snapshot of a spherical region and some of the rings from A. One can easily appreciate the interpenetration between the rings in the system (for instance, cyan through dark gray or orange through blue). The sketched network in the top corner is obtained by visual inspection and depicts the penetrations of the rings. Two circles are connected by a directed arrow when the former is threading through the latter (13) (see also Movie S1). (C) Lattice animal representation of a single ring polymer surrounded by nonthreading rings (black dots) and penetrated by threading rings (colored dots). Adapted from ref. 6 with permission from Macmillan Publishers Ltd: Nature Materials, copyright 2008.