  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.
