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