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We consider non-Hermitian energy band theory in two-dimensional systems, andstudy eigenenergy braids on slices in the two-dimensional Brillouin zone. Weshow the consequences of reciprocity and geometric symmetry on such eigenenergybraids. The point-gap topology of the energy bands can be found from theprojection of the eigenenergy braid onto the complex energy plane. We show thatthe conjugacy class transitions in the eigenenergy braid results in the changesin the number of bands in a complete point-gap loop. This transition occurs atexceptional points. We numerically demonstrate these concepts usingtwo-dimensional reciprocal and nonreciprocal photonic crystals.

Eigenenergy braids in 2D photonic crystals