  We derive an analytic formula of genus for dark matter halos assuming that
the primordial mass density field obeys the random-Gaussian statistics. In
particular, we for the first time take account of the nonlinear nature of the
halo biasing, in addition to the nonlinear gravitational evolution of the
underlying mass distribution. We examine in detail the extent to which the
predicted genus for halos depends on the redshift, smoothing scale of the
density field, and the mass of those halos in representative cold dark matter
models. In addition to the full model predictions, we derive an explicit
perturbation formula for the halo genus which can be applied to a wider class
of biasing model for dark matter halos. With the empirical relation between the
halo mass and the luminosity/temperature of galaxy clusters, our prediction can
be compared with the upcoming cluster redshift catalogues in optical and X-ray
bands. Furthermore, we discuss the detectability of the biasing and
cosmological model dependence from the future data.
