Based on a general transport theory for non-reciprocal non-Hermitian systems and a topological model that encompasses a wide range of previously studied models, we (i) provide conditions for effects such as reflectionless and transparent transport, lasing, and coherent perfect absorption, (ii) identify which effects are compatible and linked with each other, and (iii) determine by which levers they can be tuned independently. For instance, the directed amplification inherent in the non-Hermitian skin effect does not enter the spectral conditions for reflectionless transport, lasing, or coherent perfect absorption, but allows to adjust the transparency of the system. In addition, in the topological model the conditions for reflectionless transport depend on the topological phase, but those for coherent perfect absorption do not. This then allows us to establish a number of distinct transport signatures of non-Hermitian, nonreciprocal, and topological behaviour, in particular (I) reflectionless transport in a direction that depends on the topological phase, (II) invisibility coinciding with the skin-effect phase transition of topological edge states, and (III) coherent perfect absorption in a system that is transparent when probed from one side.
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