Eigenvalue topology of non-Hermitian band structures in two and three dimensions

In the band theory for non-Hermitian systems, energy eigenvalues, which are complex, can exhibit non-trivial topology is not present in Hermitian systems. one dimension, it was recently noted theoretically and demonstrated experimentally that eigenvalue classified by braid group. The classification of higher dimensions, however, remained an open question. Here, we give a complete description two three dimensional including gapped gapless cases. We reduce topological problem to purely computational algebraic topology. Brillouin zone torus punctured exceptional points, each nontrivial loop acquires group invariant. These braids satisfy constraint composite around points equal commutator on fundamental cycles torus. there knots links, depends how they embedded three-torus. When link contained contractible ball, be expressed terms knot link. Our results provide comprehensive understanding should important further explorations topologically robust quantum classical