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September  2017, 37(9): 4835-4856. doi: 10.3934/dcds.2017208

Connected components of meanders: Ⅰ. bi-rainbow meanders

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".

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.

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
References:
[1]

V. I. Arnol'd, A branched covering of $CP^2 \to S^4$, hyperbolicity and projectivity topology, Siberian Mathematical Journal, 29 (1988), 717-726.  doi: 10.1007/BF00970265.  Google Scholar

[2]

S. Cautis and D. M. Jackson, The matrix of chromatic joins and the Temperley-Lieb algebra, Journal of Combinatorial Theory. Series B, 89 (2003), 109-155.  doi: 10.1016/S0095-8956(03)00071-6.  Google Scholar

[3]

V. Coll, C. Magnant and H. Wang, The signature of a meander, preprint, arXiv: 1206.2705. Google Scholar

[4]

V. Dergachev and A. Kirillov, Index of Lie algebras of seaweed type, Journal of Lie Theory, 10 (2000), 331-343.   Google Scholar

[5]

P. Di FrancescoO. Golinelli and E. Guitter, Meanders and the Temperley-Lieb algebra, Communications in Mathematical Physics, 186 (1997), 1-59.  doi: 10.1007/BF02885671.  Google Scholar

[6]

B. Fiedler and P. Castañeda, Rainbow meanders and Cartesian billiards, São Paulo Journal of Mathematical Sciences, 6 (2012), 247-275.  doi: 10.11606/issn.2316-9028.v6i2p247-275.  Google Scholar

[7]

B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, Journal of Differential Equations, 125 (1996), 239-281.  doi: 10.1006/jdeq.1996.0031.  Google Scholar

[8]

B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, Journal of Differential Equations, 156 (1999), 282-308.  doi: 10.1006/jdeq.1998.3532.  Google Scholar

[9]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. Ⅱ: Connection graphs, Journal of Differential Equations, 244 (2008), 1255-1286.  doi: 10.1016/j.jde.2007.09.015.  Google Scholar

[10]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type.Ⅰ: Bipolar orientations and Hamiltonian paths, Journal für die Reine und Angewandte Mathematik, 635 (2009), 71-96.  doi: 10.1515/CRELLE.2009.076.  Google Scholar

[11]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. Ⅲ: Small and platonic examples, Journal of Dynamics and Differential Equations, 22 (2010), 121-162.  doi: 10.1007/s10884-009-9149-2.  Google Scholar

[12]

A. Karnauhova, Meanders: Sturm Global Attractors, Seaweed Lie Algebras and Classical Yang–Baxter Equation, 1st edition, Walter de Gruyter GmbH, Berlin/ Boston, 2017. doi: 10.1515/9783110533026.  Google Scholar

[13]

M. La Croix, Approaches to the Enumerative Theory of Meanders, Master's thesis, 2003, http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf. Google Scholar

[14]

C. Rocha, Properties of the attractor of a scalar parabolic PDE, Journal of Dynamics and Differential Equations, 3 (1991), 575-591.  doi: 10.1007/BF01049100.  Google Scholar

[15]

H. V. N. Temperley and E. H. Lieb, Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 322 (1971), 251-280.  doi: 10.1098/rspa.1971.0067.  Google Scholar

[16]

B. W. Westbury, The representation theory of the Temperley-Lieb algebras, Mathematische Zeitschrift, 219 (1995), 539-565.  doi: 10.1007/BF02572380.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, A branched covering of $CP^2 \to S^4$, hyperbolicity and projectivity topology, Siberian Mathematical Journal, 29 (1988), 717-726.  doi: 10.1007/BF00970265.  Google Scholar

[2]

S. Cautis and D. M. Jackson, The matrix of chromatic joins and the Temperley-Lieb algebra, Journal of Combinatorial Theory. Series B, 89 (2003), 109-155.  doi: 10.1016/S0095-8956(03)00071-6.  Google Scholar

[3]

V. Coll, C. Magnant and H. Wang, The signature of a meander, preprint, arXiv: 1206.2705. Google Scholar

[4]

V. Dergachev and A. Kirillov, Index of Lie algebras of seaweed type, Journal of Lie Theory, 10 (2000), 331-343.   Google Scholar

[5]

P. Di FrancescoO. Golinelli and E. Guitter, Meanders and the Temperley-Lieb algebra, Communications in Mathematical Physics, 186 (1997), 1-59.  doi: 10.1007/BF02885671.  Google Scholar

[6]

B. Fiedler and P. Castañeda, Rainbow meanders and Cartesian billiards, São Paulo Journal of Mathematical Sciences, 6 (2012), 247-275.  doi: 10.11606/issn.2316-9028.v6i2p247-275.  Google Scholar

[7]

B. Fiedler and C. Rocha, Heteroclinic orbits of semilinear parabolic equations, Journal of Differential Equations, 125 (1996), 239-281.  doi: 10.1006/jdeq.1996.0031.  Google Scholar

[8]

B. Fiedler and C. Rocha, Realization of meander permutations by boundary value problems, Journal of Differential Equations, 156 (1999), 282-308.  doi: 10.1006/jdeq.1998.3532.  Google Scholar

[9]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. Ⅱ: Connection graphs, Journal of Differential Equations, 244 (2008), 1255-1286.  doi: 10.1016/j.jde.2007.09.015.  Google Scholar

[10]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type.Ⅰ: Bipolar orientations and Hamiltonian paths, Journal für die Reine und Angewandte Mathematik, 635 (2009), 71-96.  doi: 10.1515/CRELLE.2009.076.  Google Scholar

[11]

B. Fiedler and C. Rocha, Connectivity and design of planar global attractors of Sturm type. Ⅲ: Small and platonic examples, Journal of Dynamics and Differential Equations, 22 (2010), 121-162.  doi: 10.1007/s10884-009-9149-2.  Google Scholar

[12]

A. Karnauhova, Meanders: Sturm Global Attractors, Seaweed Lie Algebras and Classical Yang–Baxter Equation, 1st edition, Walter de Gruyter GmbH, Berlin/ Boston, 2017. doi: 10.1515/9783110533026.  Google Scholar

[13]

M. La Croix, Approaches to the Enumerative Theory of Meanders, Master's thesis, 2003, http://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf. Google Scholar

[14]

C. Rocha, Properties of the attractor of a scalar parabolic PDE, Journal of Dynamics and Differential Equations, 3 (1991), 575-591.  doi: 10.1007/BF01049100.  Google Scholar

[15]

H. V. N. Temperley and E. H. Lieb, Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 322 (1971), 251-280.  doi: 10.1098/rspa.1971.0067.  Google Scholar

[16]

B. W. Westbury, The representation theory of the Temperley-Lieb algebras, Mathematische Zeitschrift, 219 (1995), 539-565.  doi: 10.1007/BF02572380.  Google Scholar

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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