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 or curve 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 and needs to be determined.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 up to four 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. In fact, the number of connected components of bi-rainbow meanders with more than four families cannot be expressed as the gcd of polynomials in the sizes of the rainbows.On the other hand, we provide a complexity analysis of nose-retraction algorithms. They determine the number of connected components of arbitrary birainbow meanders in logarithmic time. In fact, the nose-retraction algorithms resemble the Euclidean algorithm, which is used to determine the gcd, in structure and complexity.Looking for a closed formula of the number of connected components, the noseretraction algorithm is as good as a gcd-formula and therefore as good as we can possibly expect. Meander curves

Meander Graphs and Frobenius Seaweed Lie Algebras III

“…For maximal parabolics in the Type-A case, this was done early on by Elashvili, and later, as a corollary to their combinatorial result, by Dergachev and A. Kirillov, and still later by Coll et al, as a consequence of a more generalized formula. The Type-C case was addressed by Coll et al in [10]. Investigations by Karnauhova and Liebsher [13] show that these formulas are the full complement of linear greatest common divisor index formulas available in the Type-A and Type-C cases.…”

“…Consequently, the construction of Type-C meanders and related index formulas for Type-C seaweeds [10,5], as well as results on the homotopy type of Type-C meanders, carry over Mutatas Mutandis to the Type-B case.…”

“…In [7,8], Coll et al used the signature to develop closed form formulas for the index of a seaweed algebra in terms of the block sizes of the defining flags in the Type-A and Type-C cases when the number of blocks in the flags is small. Subsequently, Karnauhova and Liebsher [10] used signature type moves and complexity arguments to establish that the index formulas developed in these papers are the only linear greatest common divisor formulas for the index based on the flags defining the seaweed. One finds that in the Type-A case, a seaweed is Frobenius precisely when its associated meander consists of a single path [4,7,8].…”

“…One finds that in the Type-A case, a seaweed is Frobenius precisely when its associated meander consists of a single path [4,7,8]. For a Type-C seaweed to be Frobenius, its associated meander must reduce to a certain collection of paths [5,10].…”

Symplectic meanders

“…More importantly, the signature can be used to test any relatively prime conditions which might serve to identify a Frobenius seaweed subalgebra of sl(n). Indeed, using signature moves and complexity arguments, Karnauhova and Liebscher [9] show, in particular, that there is no linear gcd formula for finding the index of the general seaweed p((a 1 , . .…”