  We simulate CMB maps including non-Gaussianity arising from cubic order
perturbations of the primordial gravitational potential, characterized by the
non-linearity parameter $g_{NL}$. The maps are used to study the characteristic
nature of the resulting non-Gaussian temperature fluctuations. We measure the
genus and investigate how it deviates from Gaussian shape as a function of
$g_{NL}$ and smoothing scale. We find that the deviation of the non-Gaussian
genus curve from the Gaussian one has an antisymmetric, sine function like
shape, implying more hot and more cold spots for $g_{NL}>0$ and less of both
for $g_{NL}<0$. The deviation increases linearly with $g_{NL}$ and also
exhibits mild increase as the smoothing scale increases. We further study other
statistics derived from the genus, namely, the number of hot spots, the number
of cold spots, combined number of hot and cold spots and the slope of the genus
curve at mean temperature fluctuation. We find that these observables carry
signatures of $g_{NL}$ that are clearly distinct from the quadratic order
perturbations, encoded in the parameter $f_{NL}$. Hence they can be very useful
tools for distinguishing not only between non-Gaussian temperature fluctuations
and Gaussian ones but also between $g_{NL}$ and $f_{NL}$ type
non-Gaussianities.
