  Fractional quantum Hall (FQH) states are topologically ordered which
indicates that their essential properties are insensitive to smooth
deformations of the manifold on which they are studied. Their microscopic
Hamiltonian description, however, strongly depends on geometrical details.
Recent work has shown how this dependence can be exploited to generate
effective models that are both interesting in their own right and also provide
further insight into the quantum Hall system. We review and expand on recent
efforts to understand the FQH system close to the solvable thin-torus limit in
terms of effective spin chains. In particular, we clarify how the difference
between the bosonic and fermionic FQH states, which is not apparent in the
thin-torus limit, can be seen at this level. Additionally, we discuss the
relation of the Haldane-Shastry chain to the so-called QH circle limit and
comment on its significance to recent entanglement studies.
