We provide an overview of Forman-Ricci curvature and persistent homology, and how their combination can be applied to the study of networks. We discuss how the usually employed augmented Forman-Ricci curvature formula, only valid for quasiconvex augmented networks, can be extended to the non-quasiconvex case. We apply three versions of quasiconvex Forman-Ricci curvature (plain, triangle-augmented, and pentagon-augmented) to build time filtrations on non-quasiconvex networks, both model and real-world. Our results suggest that triangle-augmented curvature should be used until the non-quasiconvex formula is further studied, as plain curvature omits too much information, and quasiconvex pentagon-augmented curvature is too rough of an approximation and significantly distorts the results.