  Topologically non-trivial phases have recently been reported on self-similar
structures. Here, we investigate the effect of local structure, specifically
the role of the coordination number, on the topological phases on self-similar
structures embedded in two dimensions. We study a geometry dependent model on
two self-similar structures having different coordination numbers, constructed
from the Sierpinski Gasket. For different non-spatial symmetries present in the
system, we numerically study and compare the phases on both the structures. We
characterize these phases by the localization properties of the single-particle
states, their robustness to disorder, and by using a real-space topological
index. We find that both the structures host topologically non-trivial phases
and the phase diagrams are different on the two structures. This suggests that,
in order to extend the present classification scheme of topological phases to
non-periodic structures, one should use a framework which explicitly takes the
coordination of sites into account.
