The use of interferometric nulling for the direct detection of extrasolar planets is in part limited by the extreme sensitivity of the instrumental response to tiny optical path differences between apertures. The recently proposed kernel-nuller architecture attempts to alleviate this effect with an all-in-one combiner design that enables the production of observables inherently robust to residual optical path differences (<< lambda). Until now, a unique kernel nuller design has been proposed ad hoc for a four-beam combiner. We examine the properties of this original design and generalize them for an arbitrary number of apertures. We introduce a convenient graphical representation of the complex combiner matrices that model the kernel nuller and highlight the symmetry properties that enable the formation of kernel nulls. The analytical description of the nulled outputs we provide demonstrates the properties of a kernel nuller. Our description helps outline a systematic way to build a kernel nuller for an arbitrary number of apertures. The designs for 3- and 6-input combiners are presented along with the original 4-input concept. Combiners grow in complexity with the square of the number of apertures. While one can mitigate this complexity by multiplexing nullers working independently over a smaller number of sub-apertures, an all-in-one kernel nuller recombining a large number of apertures appears as the most efficient way to characterize a high-contrast complex astrophysical scene. One can design kernel nullers for an arbitrary number of apertures that produce observable quantities robust to residual perturbations. The designs we recommend are lossless and take full advantage of all the available interferometric baselines. They are complete, result in as many kernel nulls as the theoretically expected number of closure-phases, and are optimized to require as few outputs as possible.
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