  The region of filling factors $1/3<\nu<2/5$ is predicted to support new types
of fractional quantum Hall states with topological order different from that of
the Laughlin-Jain or the Moore-Read states. Incompressibility is a necessary
condition for the formation of such novel topological states. We find that at
6.9~mK incompressibility develops only at $\nu=4/11$ and $5/13$, while the
states at $\nu=6/17$ and $3/8$ remain compressible. Our observations at
$\nu=4/11$ and $5/13$ are first steps towards understanding emergent
topological order in these fractional quantum Hall states.
