We consider non-Hermitian energy band theory in two-dimensional systems, and study eigenenergy braids on slices in the two-dimensional Brillouin zone. We show the consequences of reciprocity and geometric symmetry on such eigenenergy braids. The point-gap topology of the energy bands can be found from the projection of the eigenenergy braid onto the complex energy plane. We show that the conjugacy class transition in the eigenenergy braid results in changes in the number of bands in a complete point-gap loop. This transition occurs at exceptional points. We numerically demonstrate these concepts using two-dimensional reciprocal and nonreciprocal photonic crystals.

• Received 27 July 2023
• Revised 25 October 2023
• Accepted 26 October 2023

DOI:https://doi.org/10.1103/PhysRevB.108.195413