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 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) such n-tuples, and that all of them arise from the topology of C(f) for some polynomial f .