General decay of solutions of a Bresse system with viscoelastic boundary conditions
Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure
September 2017, 37(9): 4835-4856. doi: 10.3934/dcds.2017208
Connected components of meanders: Ⅰ. bi-rainbow meanders
Anna Karnauhova , and Stefan Liebscher
Free University Berlin, Arnimallee 3, Berlin 14195, Germany
* Corresponding author: Anna Karnauhova
Received March 2016 Revised April 2017 Published June 2017
Fund Project: Both authors are supported by the CRC 647 "Space-Time-Matter".
Figure(16)
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.
Keywords: Dynamical systems, meanders, connected components, seaweed Lie algebras, Cartesian billiard, Temperley–Lieb algebra.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Anna Karnauhova, Stefan Liebscher. Connected components of meanders: Ⅰ. bi-rainbow meanders. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4835-4856. doi: 10.3934/dcds.2017208
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Figure 2. A general meander as a pair of arc collections
Figure 1. Some examples of bi-rainbow meanders and colored connected components; a) $\mathcal{R}\mathcal{M}(1, 2, 3, 4, 5)$; b) $\mathcal{R}\mathcal{M}(1, 5, 2, 3, 4)$; c) $\mathcal{R}\mathcal{M}(4, 5, 1, 2, 3)$
Figure 3. Correspondence of cycles of $\pi^⩍\sigma$ and faces of the planar graph in Proposition 1
Figure 4. Shooting permutation of a connected meander
Figure 5. Flip of the lower arc collection of a meander
Figure 6. Generators of the Temperley-Lieb algebra $TL_{\alpha}(q)$ as strand diagrams
Figure 7. Meander as the closure of an element of a Temperley-Lieb algebra
Figure 8. Meander as trajectories of a Cartesian billiard
Figure 9. General bi-rainbow meander
Figure 10. Collapse of the bi-rainbow meander $\mathcal{R}\mathcal{M}(7, 2, 3, 5, 4)$. From top to bottom: $\mathcal{R}\mathcal{M}$, colored domains, and $\mathcal{C}\mathcal{R}\mathcal{M}$. The collapsed bi-rainbow meander consists of one path, two cycles, and an isolated point (counted as a second path)
Figure 11. Collapse for general meanders: a) general meaner; b) colored domain; c) gluing of neighboring vertices and arcs; d) general collapsed meander with branched curves
Figure 12. Outer nose retractions of bi-rainbow meanders
Figure 13. Cutting of Cartesian billiards to resemble outer nose retractions
Figure 14. Inner nose retraction of a bi-rainbow meander
Figure 15. Cutting of Cartesian billiards to resemble inner nose retractions
Figure 16. Sequence of connected meanders generated from $\mathcal{R}\mathcal{M}(1, 1, 2)$ by iterated inverse outer nose retractions
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