Persistent Homology of Entangled Rings

Abstract

Long polymer chains are subject to topological constraints, which often dictate the physical properties of the system. Here we propose an unambiguous algorithm to characterize topological constraints in polymer systems based on persistent homology, a key mathematical tool to extract robust topological information from large datasets. We apply the method to the dense solution of non-concatenated entangled rings, where the ring-specific topological constraint, called threading, is often invoked in the literature, whose characterization is, however, nontrivial. Our formulation provides a natural definition of the ring’s threading into another, whereby allowing us to analyze the statistics of threading events in the system. The method is not restricted to ring polymers, thus expected to find broader applications for the study of topological constraints in generic polymeric materials.