  We present an extensive quantum Monte Carlo study of a nearest-neighbor,
singlet-projection model on the kagome lattice that exhibits SO($N$) symmetry
and is sign-problem-free. We find that in contrast to the previously studied
SU($N$) variations of this model, the non-bipartite lattices appear to harbor
spin-liquid phases for intermediate values of $N$, a result also seen on the
triangular lattice with this same model. Unlike the triangular lattice,
however, the kagome lattice appears to admit no valence bond solid (VBS) phase
for large $N$, instead remaining a spin-liquid. We also observe that the
spin-ordered phase survives to a relatively large value of $N$, at least $N=8$,
and that it is gone for $N=10$; the fate of $N=9$ remains unclear.
