Quantum Hall States at filling $\nu=\frac{2}{k+2}$

We study the $\nu=\frac{2}{k+2}$ quantum Hall states which are particle-hole conjugates of the $\nu=\frac{k}{k+2}$ Read-Rezayi states. We find that equilibration between the different modes at the edge of such a state leads to an emergent SU(2)$_k$ algebra in the counter-propagating neutral sector. Heat flow along the edges of these states will be in the opposite direction of charge flow. In the $k=3$ case, which may be relevant to $\nu=2+\frac{2}{5}$, the thermal Hall conductance and the exponents associated with quasiparticle and electron tunneling distinguish this state from competing states such as the hierarchy/Jain state.