Meander determinants (Q1270121)

A meander is a closed non-self-intersecting loop crossing an infinite line through \(2n\) points, with two being considered equivalent if they are smooth deformations of each other. Picture a closed circuit crossing a river over \(2n\) bridges. A semi-meander replaces the line with a half-line (so a river with a source which can be walked around). Cutting a meander along the river results in two arch configurations, one above the river and one below, each with \(n\) arches. A similar cutting of a semi-meander also requires cutting the paths around the source and extending the line segments to half-lines. The result is two open arch configurations each consisting of \(n\) points on a line, \(h\) of which have half-lines extending from them and \(n-h\) connected in pairs by arches. The semi-meander is reconstructed from the two open arch configurations by identifying the \(n\) points along the half-line and then connecting the \(h\) open lines from right to left around the rivers source (this river flows from east to west on your map). Simple recursive and closed formulas are found for the number of arch and open arch configurations, \(c_n\) and \(c_{n,h}\). Given two arch or semi-arch configurations, \(a\) and \(b\), they can be attached to form a multi-component meander or semi-meander. Let \(\kappa(a| b)\) be the number of connected components formed when connecting \(a\) and \(b\). Given a complex number, \(q\), the meander and semi-meander matrices are defined by \([{\mathcal G}_{2n}(q)]_{a,b}=q^{\kappa(a| b)}\) and \([{\mathcal G}_{2n}^{(h)}(q)]_{a,b}=q^{\kappa(a| b)}\). The two main theorems of this paper give formulas for the determinants of the meander and semi-meander matrices in terms of Chebyshev polynomials of the second kind. The formula for the semi-meander case is new and generalizes the formula for the meander determinant. The road to these results is as interesting as the destination. Meanders are identified with walk diagrams and with the product of two elements in the Temperley-Lieb algebra. The result is obtained by reformulating the meander determinant as the Gram determinant of a particular basis of the Temperley-Lieb algebra. The Gram-Schmidt orthogonalization is then carried out on this basis. With the meander case as a model, a similar strategy is carried out for the semi-meander case with some complications.