Year: 2026
Type: preprint
Abstract: A meander is a configuration of two simple plane curves intersecting transversely. The orders of their intersection points define a permutation that determines the configuration. When tangencies are allowed, however, different configurations can share the same permutation. We study the combinatorial and algebraic structures arising from this non-uniqueness. We give a realization criterion and show that the realizations of each realizable permutation form an affine space over the two-element field. We describe this space using an associated graph, called the component spine. We prove that the component spine of every permutation is a cactus. We also introduce the tangency polynomial, which counts realizations by their number of tangencies, investigate its properties, and prove that it factors over the cycles and bridges of the component spine. We derive a central limit theorem for tangency counts and obtain asymptotic formulas for the number of distinct tangency polynomials.
Source: arXiv (Cornell University)
Authors: Belousov, Yury, Georgievskaia, Viktoriia
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Award: G4466472670
OpenAlex ID: W7214375808