Many rings with long lifespans then form over the time-span 40.0 µs to 60.0 µs with few rings formed in the final 60.0 µs to 100 µs as the system is A magnified view of the ring life lines is shown in Figure and emphasises a key difference this approach has from the traditional simplicial complex approach used in persistent homology. As discussed in Section 1, a dynamical system varying over time there is no guarantee that a given ring R that is born time t will still exist at a later time t ′ . Furthermore, dying at time t ′ does not preclude a ring from being born again even later at time t ′′ . The loss of the mathematical continuity of simplices prevents us from using some further persistent homology analysis techniques (Stolz-Pretzer ( )), such as using the barcodes to measure similarity of trajectories as performed by or to identify spatially important information such as binding sites in the manner of . However, it is useful to identify showing detail on the life-lines of some rings are dead for extended periods of time but are reborn and continue to exist afterwards. For the higher density network shown in detail in Figure (b) we see a key difference from the low density network. For a high density network, the rings are close to one another and the polymer edges are packed in tightly. This means that few rings with long continuous lifespans are observed, but many microscopic damage / repair events happen continuously. This can be quantified with the data in Table , showing there are more unique hexagons observed for as the area per molecule decreases and the hexagons that are observed are reborn more often.