Disordered carbons comprise graphene fragments assembled into three-dimensional networks. It has long been debated whether these networks contain positive curvature, as seen in fullerenes, negative curvature, as proposed for the schwarzite structures, or zero curvature, as in ribbons. We present a mesh-based approach to analyze the topology of a set of nanoporous and glassy carbon models that accurately reproduce experimental properties. Although all three topological elements are present, negatively curved structures dominate. At the atomic level, analysis of local environments shows that sp- and sp^{3}-bonded atoms are associated with line defects and screw dislocations that resolve topological complexities such as termination of free edges and stacking of low curvature regions into ribbons. These results provide insight into the synthesis of porous carbon materials, glassy carbon and the graphitizability of carbon materials.