  For every polynomial f of degree n with no double roots, there is an
associated family C(f) of harmonic algebraic curves, fibred over the circle,
with at most n-1 singular fibres. We study the combinatorial topology of C(f)
in the generic case when there are exactly n-1 singular fibres. In this case,
the topology of C(f) is determined by the data of an n-tuple of noncrossing
matchings on the set {0,1,...,2n-1} with certain extra properties. We prove
that there are 2(2n)^{n-2} such n-tuples, and that all of them arise from the
topology of C(f) for some polynomial f.
