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The electronic properties of $\pi$-conjugated two-dimensional (2D) polymersnear the Fermi level are determined by structural topology and chemicalcomposition. Thus, tight-binding (TB) calculations of the correspondingfundamental network can be used to explore the parameter space to findconfigurations with intriguing properties before designing the the atomistic 2Dpolymer network. The vertex-transitive \textbf{fes} lattice, which is alsocalled square-octagon lattice, is rich in interesting topological featuresincluding Dirac points and flat bands. Herein, we study its electronic andtopological properties within the TB framework using representative parametersfor chemical systems. Secondly, we demonstrate that the rational implementationof band structure features obtained from TB calculations into 2D polymers isfeasible with a family of 2D polymers possessing \textbf{fes} structure. Aone-to-one band structure correspondence between fundamental network and 2Dpolymers is found. Moreover, changing the relative length of linkers connectingthe triangulene units in the 2D polymers reflect tuning of hopping parametersin the TB model. These perturbations allow to open sizeable local band gaps atvarious positions in the Brillouin zone. From analysis of Berry curvature flux,none of the polymers exhibits a large topologically non-trivial band gap.However, we find a particular configuration of semimetallic characteristicswith separate electron and hole pockets, which possess very low effectivemasses both for electrons (as small as $m^*_\mathrm{e} = 0.05$) and holes (assmall as $m^*_\mathrm{h} = 0.01$).

A semimetallic square-octagon (fes) two-dimensional polymer with high mobility