  Two-dimensional electron dispersions with peculiar band crossings provide a
platform for realizing topological phases of matter. Here we theoretically show
that the $e_g$-orbital manifold of honeycomb-layered transition metal compounds
accommodates a plethora of peculiar band crossings, such as multiple Dirac
point nodes, quadratic band crossings, and line nodes. From the tight-binding
analysis, we find that the band topology is systematically changed by the
orbital dependent transfer integrals on the honeycomb network of edge-sharing
octahedra, which can be modulated by distortions of the octahedra as well as
chemical substitutions. The band crossings are gapped out by the spin-orbit
coupling, which brings about a variety of topological phases distinguished by
the spin Chern numbers. The results provide a comprehensive understanding of
the previous studies on various honeycomb compounds. We also propose another
candidate materials by ab initio calculations.
