  By introducing the transformation optics method to periodic systems, we show
the tunability of the band structures by comparing the results from original
spaces and transformed spaces. Interestingly, we find the topological invariant
Chern number will change sign when the orientation of the Brillouin zone
flipped. The new platform we provided for engineering the band diagram and
topological invariant might lead to the development of both transformation
optics and photonic topological states.
