On meandric permutations

TL;DR

This paper establishes a bijection between meanders and special Gauss diagrams derived from braid group generators, providing algorithms and matrix criteria for analyzing meandric permutations.

Contribution

It introduces a novel bijection and matrix coding method for meanders, linking topological and algebraic structures in a new way.

Findings

Constructs an algorithm to generate Gauss diagrams from meanders.

Shows matrices associated with diagrams are idempotent over GF(2).

Provides a criterion to determine if a permutation is meandric.

Abstract

We give a bijection between meanders and special sorts of Gauss diagrams which aroused from the Thurston generators of braid groups. It allows us to give an algorithm to construct such diagrams and to code meanders by matrices which are exactly incident matrices of the corresponding adjacency graphs of the diagrams. Finally, we show that these matrices are idempotent over the field $\mathsf{GF}(2)$, and obtain a criterion for a permutation to be meandric.