  We study the possibility of fractional quantum Hall effects in HgTe quantum
wells using exact diagonalization. Our results show that Laughlin states, the
Moore-Read state, and the Read-Rezayi $Z_3$ state can all be supported.
However, near the level crossing point (of the single-particle spectrum), the
gap can be destroyed by Landau level mixing, and the Moore-Read state and the
Read-Rezayi state dominate over their respective competing states only for wide
wells. For smaller well widths the Moore-Read state crosses over to the
composite fermion Fermi sea, while the Read-Rezayi state loses its dominance
over the hierarchy state.
