  (Abridged) We use the triaxial halo model of Smith & Watts to investigate the
impact of dark matter halo shapes on the bispectrum. Analytic expressions are
derived and evaluated through numerical integration. Two models for the
ellipsoidal halo profiles are considered: a toy model designed to isolate the
effects of halo shape on the clustering alone; and the realistic model of Jing
& Suto. For equilateral k-space triangles, we show that the predictions of the
triaxial model are suppressed, relative to the spherical model, by roughly ~7%
and ~4% for the two profiles respectively. The reduced bispectrum as a function
of triangle configuration is found to be very sensitive to halo shapes on small
scales. The generic features are that, relative to the spherical model, the
signal is suppressed for k-space triangles that are close to equilateral and
boosted for configurations that are colinear. This appears to be a unique
signature of halo triaxiality.
  The galaxy bispectrum is also explored. Two models for the halo occupation
distribution (HOD) are considered: the binomial distribution of Scoccimarro et
al. and the Poisson satellite model of Kravtsov et al. Our predictions show
that the galaxy bispectrum is also sensitive to halo shapes, although relative
to the mass, the effects are reduced. The HOD of Kravtsov et al. is found to be
more sensitive. This owes to the mean number of galaxies per halo being a
steeper function of mass in this model, hence giving stronger weight to the
high mass (more triaxial) haloes. Interestingly, the functional form is not
strongly sensitive to the details of the HOD, but is mainly determined by the
halo shape. However, a combination of measurements made on different scales and
for different k-space triangles are sensitive to both halo shape and the HOD.
