  One-dimensional superlattices with periodic spatial modulations of onsite
potentials or tunneling coefficients can exhibit a variety of properties
associated with topology or symmetry. Recent developments of ring-shaped
optical lattices allow a systematic study of those properties in superlattices
with or without boundaries. While superlattices with additional modulating
parameters are shown to have quantized topological invariants in the augmented
parameter space, we also found localized or zero-energy states associated with
symmetries of the Hamiltonians. Probing those states in ultracold-atoms is
possible by utilizing recently proposed methods analyzing particle depletion or
the local density of states. Moreover, we summarize feasible realizations of
configurable optical superlattices using currently available techniques.
