  We show that non-Hermiticity enables topological phases with unidirectional
transport in one-dimensional Floquet chains. The topological signatures of
these phases are non-contractible loops in the spectrum of the Floquet
propagator that are separated by an imaginary gap. Such loops occur exclusively
in non-Hermitian Floquet systems. We define the corresponding topological
invariant as the winding number of the Floquet propagator relative to the
imaginary gap. To relate topology to transport, we introduce the concept of
regularized dynamics of non-Hermitian chains. We establish that, under the
conditions of regularized dynamics, transport is quantized in so far as the
charge transferred over one period equals the topological winding number. We
illustrate these theoretical findings with the example of a Floquet chain that
features a topological phase transition and acts as a charge pump in the
non-trivial topological phase. We finally discuss whether these findings
justify the notion that non-Hermitian Floquet chains support topological
transport.
