Check Inversion Symmetry: Honeycomb Kitaev Model

[MatNiss]

Homework Statement

Hey,
I need to check if a specific lattice configuration has a inversion symmetry. The lattice structure shows a specific vortex configuration in the honeycomb lattice using the Kitaev model. The black cells means that the plaquette contains a vortex and a white cell means that there is no vortex. The white dot means that the site belongs to the even and a black dot that it belongs to the odd sub lattice. The green line implies that the link between the site is a minus link, all the other links are positive. The inversion symmetry should be checked here by applying the inversion operator P which projects each site to
c_j \to c_{-j}
And it is written in the literature, that a Gauge transformation need to be applied. Can someone maybe explain how this gauge transformation works, how I exactly perform the inversion?

Relevant Equations

P=GI
I: c_j \to c_{-j}
G: c_j \to g_j c_{-j}

 

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