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

Abstract

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