Polynomials, meanders, and paths in the lattice of noncrossing partitions

For every polynomial $f$ of degree $n$ with no double roots, there is an associated family $\mathcal{C}(f)$ of harmonic algebraic curves, fibred over the circle, with at most $n-1$ singular fibres. We study the combinatorial topology of  $\mathcal{C}(f)$ in the generic case when there are exactly $n-1$ singular fibres. In this case, the topology of  $\mathcal{C}(f)$ is determined by the data of an $n$-tuple of noncrossing matchings on the set $\{0,1,\ldots,2n-1\}$ 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 $\mathcal{C}(f)$ for some polynomial $f$.