  We consider an annular superconductor-insulator-superconductor
Josephson-junction, with the insulator being a double layer of electron and
holes at Abelian fractional quantum Hall states of identical fillings. When the
two superconductors gap out the edge modes, the system has a topological ground
state degeneracy in the thermodynamic limit akin to the fractional quantum Hall
degeneracy on a torus. In the quasi-one-dimensional limit, where the width of
the insulator becomes small, the ground state energies are split. We discuss
several implications of the topological degeneracy that survive the crossover
to the quasi-one-dimensional limit. In particular, the Josephson effect shows a
$2\pi d$-periodicity, where $d$ is the ground state degeneracy in the 2
dimensional limit. We find that at special values of the relative phase between
the two superconductors there are protected crossing points in which the
degeneracy is not completely lifted. These features occur also if the insulator
is a time-reversal-invariant fractional topological insulator. We describe the
latter using a construction based on coupled wires. Furthermore, when the
superconductors are replaced by systems with an appropriate magnetic order that
gap the edges via a spin-flipping backscattering, the Josephson effect is
replaced by a spin Josephson effect.
