  Motivated by the observation of positive Gaussian curvature in kinetoplast
DNA networks, we consider the effect of linking chirality in square lattice
molecular chainmail networks using Langevin dynamics simulations and
constrained gradient optimization. Linking chirality here refers to ordering of
over-under versus under-over linkages between a loop and its neighbors. We
consider fully alternating linking, maximally non-alternating, and partially
non-alternating linking chiralities. We find that in simulations of polymer
chainmail networks, the linking chirality dictates the sign of the Gaussian
curvature of the final state of the chainmail membranes. Alternating networks
have positive Gaussian curvature, similar to what is observed in kinetoplast
DNA networks. Maximally non-alternating networks form isotropic membranes with
negative Gaussian curvature. Partially non-alternating networks form flat
diamond-shaped sheets which undergo a thermal folding transition when
sufficiently large, similar to the crumpling transition in tethered membranes.
We further investigate this topology-curvature relationship on geometric
grounds by considering the tightest possible configurations and the constraints
that must be satisfied to achieve them.
