Connected components of meanders: Ⅰ. bi-rainbow meanders

Abstract

Closed meanders are planar configurations of one or several disjoint closed Jordan curves intersecting a given line transversely. They arise as shooting curves of parabolic PDEs in one space dimension, as trajectories of Cartesian billiards, and as representations of elements of Temperley-Lieb algebras.Given the configuration of intersections, for example as a permutation or an arc collection, the number of Jordan curves is unknown. We address this question in the special case of bi-rainbow meanders, which are given as non-branched families (rainbows) of nested arcs. Easily obtainable results for small bi-rainbow meanders containing at most four families in total (lower and upper rainbow families) suggest an expression of the number of curves by the greatest common divisor (gcd) of polynomials in the sizes of the rainbow families.We prove however, that this is not the case.On the other hand, we provide a complexity analysis of nose-retraction algorithms. They determine the number of connected components of arbitrary bi-rainbow meanders in logarithmic time. In fact, the nose-retraction algorithms resemble the Euclidean algorithm.Looking for a closed formula of the number of connected components, the nose-retraction algorithm is as good as a gcd-formula and therefore as good as we can possibly expect.