  Analyzing phase transitions using the inherent geometrical attributes of a
system has garnered enormous interest over the past few decades. The usual
candidate often used for investigation is graphene -- the most celebrated
material among the family of tri co-ordinated graphed lattices. We show in this
report that other inhabitants of the family demonstrate equally admirable
structural and functional properties that at its core are controlled by their
topology. Two interesting members of the family are Cylooctatrene(COT) and
COT-based polymer: poly-bi-[8]-annulenylene both in one and two dimensions that
have been investigated by polymer chemists over a period of 50 years for its
possible application in batteries exploiting its conducting properties. A
single COT unit is demonstrated herein to exhibit topological solitons at sites
of a broken bond similar to an open one-dimensional Su-Schrieffer-Heeger (SSH)
chain. We observe that Poly-bi-[8]-annulenylene in 1D mimics two coupled SSH
chains in the weak coupling limit thereby showing the presence of topological
edge modes. In the strong coupling limit, we investigate the different
parameter values of our system for which we observe zero energy modes. Further,
the application of an external magnetic field and its effects on the
band-flattening of the energy bands has also been studied. In 2D,
poly-bi-[8]-annulenylene forms a square-octagon lattice which upon breaking
time-reversal symmetry goes into a topological phase forming noise-resilient
edge modes. We hope our analysis would pave the way for synthesizing such
topological materials and exploiting their properties for promising
applications in optoelectronics, photovoltaics, and renewable energy sources.
