Topology of Large-Scale Structures of Galaxies in Two Dimensions -
  Systematic Effects
  We study the two-dimensional topology of the galactic distribution when
projected onto two-dimensional spherical shells. Using the latest Horizon Run 4
simulation data, we construct the genus of the two-dimensional field and
consider how this statistic is affected by late-time nonlinear effects --
principally gravitational collapse and redshift space distortion (RSD). We also
consider systematic and numerical artifacts such as shot noise, galaxy bias,
and finite pixel effects. We model the systematics using a Hermite polynomial
expansion and perform a comprehensive analysis of known effects on the
two-dimensional genus, with a view toward using the statistic for cosmological
parameter estimation. We find that the finite pixel effect is dominated by an
amplitude drop and can be made less than $1\%$ by adopting pixels smaller than
$1/3$ of the angular smoothing length. Nonlinear gravitational evolution
introduces time-dependent coefficients of the zeroth, first, and second Hermite
polynomials, but the genus amplitude changes by less than $1\%$ between $z=1$
and $z=0$ for smoothing scales $R_{\rm G} > 9 {\rm Mpc/h}$. Non-zero terms are
measured up to third order in the Hermite polynomial expansion when studying
RSD. Differences in shapes of the genus curves in real and redshift space are
small when we adopt thick redshift shells, but the amplitude change remains a
significant $\sim {\cal O}(10\%)$ effect. The combined effects of galaxy
biasing and shot noise produce systematic effects up to the second Hermite
polynomial. It is shown that, when sampling, the use of galaxy mass cuts
significantly reduces the effect of shot noise relative to random sampling.
