  Higher-order topological phases feature topologically protected boundary
states in lower dimensions. Specifically, the zero-dimensional corner states
are protected by the $d$th-order topology of a $d$-dimension system. In this
work, we propose to predict different configurations of corner states from
winding numbers defined for one-dimensional edges of the system. We first
demonstrate the winding number characterization with a generalized
two-dimensional square lattice belonging to the BDI symmetry class. In addition
to the second-order topological insulating phase, the system may also be a
nodal point semimetal or a weak topological insulator with topologically
protected one-dimensional edge states coexisting with the corner states at zero
energy. A three-dimensional cubic lattice with richer configurations of corner
states is also studied. We further discuss several experimental implementations
of our models with photonic lattices or electric circuits.
