  We consider manifestations of topological order in time-reversal-symmetric
fractional topological liquids (TRS-FTLs), defined on planar surfaces with
holes. We derive a formula for the topological ground state degeneracy of such
a TRS-FTL, which applies to cases where the edge modes on each boundary are
fully gapped by appropriate backscattering terms. The degeneracy is exact in
the limit of infinite system size, and is given by $q^{N^{\,}_{\mathrm{h}}}$,
where $N^{\,}_{\mathrm{h}}$ is the number of holes and $q$ is an integer that
is determined by the topological field theory. When the degeneracy is lifted by
finite-size effects, the holes realize a system of $N^{\,}_{\mathrm{h}}$
coupled spin-like $q$-state degrees of freedom. In particular, we provide
examples where $\mathbb Z^{\,}_{q}$ quantum clock models are realized on the
low-energy manifold of states. We also investigate the possibility of measuring
the topological ground state degeneracy with calorimetry, and briefly revisit
the notion of topological order in $s$-wave BCS superconductors.
