Polynomials, meanders, and paths in the lattice of noncrossing partitions
Abstract
Description
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.
24 pages, 7 figures. To appear, Transactions of the A.M.S. Revised based on referee report; final section added
24 pages, 7 figures. To appear, Transactions of the A.M.S. Revised based on referee report; final section added