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.

The main changes are that Landau level mixing destroys the fractional quantum Hall effect near the level crossing point, and that the Moore-Read state and the Read-Rezayi state are favoured only for sufficiently wide quantum wells (caused by a mistake in some of the interaction matrix elements). 8 pages, 8 figures