  We have studied extensively the band crossing patterns of the bulk
entanglement spectrum (BES) for various lattice Chern insulators. We find that
only partitions with dual symmetry can have either stable nodal-lines or
nodal-points in the BES when the system is in the topological phase of a
nonzero Chern number. By deforming the Hamiltonian to lift the accidental
symmetry, one can see that only nodal points are robust. They thus should bear
certain topological characteristics of the BES. By studying the band crossing
patterns in details we conclude that the topological characteristics of the BES
are inherited from the topological order of the underlying Chern insulators and
the former can have more refined topological structures. We then propose the
conjecture that the sum of the vorticities in the BES in a properly chosen
reduced Brillouin zone equals the Chern number of the underlying Chern
insulator. This relation is beyond the usual classification scheme of
topological insulators/superconductors.
