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.