  Triangular and honeycomb lattices are dual to each other -- if we puncture
holes into a featureless plane in a regular triangular alignment, the remaining
body looks like a honeycomb lattice, and vice versa, if the holes are in a
regular honeycomb alignment, the remaining body has a feature of triangular
lattice. In this work, we reveal that the electronic states in graphene sheets
with nano-sized holes in triangular and honeycomb alignments are also dual to
each other in a topological sense. Namely, a regular hole array perforated in
graphene can open a band gap in the energy-momentum dispersion of relativistic
electrons in the pristine graphene, and the insulating states induced by
triangular and honeycomb hole arrays are distinct in topology. In a graphene
patchwork with regions of these two hole arrays put side by side
counterpropagating topological currents emerge at the domain wall. This
observation indicates that the cerebrated atomically thin sheet is where
topological physics and nanotechnology meet.
