  We show that distinct topological phases of the band structure of a
non-Hermitian Hamiltonian can be classified with elements of the braid group.
As the proof of principle, we consider the non-Hermitian evolution of the
statistics of nonequilibrium stochastic currents. We show that topologically
nontrivial phases have detectable properties, including the emergence of
decaying oscillations of parity and state probabilities, and discontinuities in
the steady state statistics of currents.
