  Even as our measurements of cosmological parameters improve, the physical
nature of the dark sector of the universe largely remains a mystery. Many
effects of dark sector models are most prominent at very large scales and will
rely on future galaxy surveys to elucidate. In this paper we compare the
topological properties of the large scale dark matter distribution in a number
of cosmological models using hydrodynamical simulations and the cosmological
genus statistic. Genus curves are computed from z = 11 to z = 0 for
{\Lambda}CDM, Quintessence and Warm Dark Matter models, over a scale range of 1
to 20 Mpc/h. The curves are analysed in terms of their Hermite spectra to
describe the power contained in non-Gaussian deformations to the cosmological
density field. We find that the {\Lambda}CDM and {\Lambda}WDM models produce
nearly identical genus curves indicating no topological differences in
structure formation. The Quintessence model, which differs solely in its
expansion history, produces significant differences in the strength and
redshift evolution of non-Gaussian modes associated with higher cluster
abundances and lower void abundances. These effects are robust to cosmic
variance and are characteristically different from those produced by tweaking
the parameters of a {\Lambda}CDM model. Given the simplicity and similarity of
the models, detecting these discrepancies represents a promising avenue for
understanding the effect of non-standard cosmologies on large-scale structure.
