A bijection between unicellular and bicellular maps

Han, Hillary S. W.; Reidys, Christian M.

doi:10.48550/arxiv.1301.7177

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…Now the dual graph G * to the arc system is a bicellular map of genus g − 1 with 2N edges: the two faces of the graph correspond to the two boundary components of the surface. The number ε [HR,Corrolary 1]. The number of meanders in that case is bounded by 2N • p g (2N + 1) • C 2N +1 , where the factor 2N accounts for the 2N possibly different identifications.…”

Section: Proofs Of the Main Results For Fixed Values Of G And Nmentioning

confidence: 99%

Enumeration of Meanders and Masur–veech Volumes

Delecroix

Goujard

Zograf

et al. 2020

Forum of Mathematics, Pi

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In memory of Maryam Mirzakhani who inspired us in so many ways.Abstract. A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with 2N crossings grows exponentially when N grows, but the longstanding problem on the precise asymptotics is still out of reach.We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as N tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator.The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinders. The proofs combine recent results on Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics. √ 145 12of the corresponding critical exponent α in a two-dimensional CFT with central charge c = −4 coupled to gravity is given in [DiFGG2].We say that a meander has a maximal arc ("rainbow " in terminology of [ACPRS]) if it has an arc joining the leftmost and the rightmost crossings with the horizontal line. Otherwise the meander does not a have maximal arc. The meander on the left of Figure 1 has a maximal arc, while the one on the right does not.By minimal arc ("pimple" in terminology of [ACPRS], or "internal arch" in terminology of [DiFGG1]) we call an arc which does not have any crossings inside.

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Section: Proofs Of the Main Results For Fixed Values Of G And Nmentioning

confidence: 99%

Enumeration of Meanders and Masur–veech Volumes

Delecroix

Goujard

Zograf

et al. 2020

Forum of Mathematics, Pi

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“…Let U 5,i g+2,n+2 denote the subset of U 5 g+2,n+2 in which only a single H γi , i = 1, 2, 3 is closed and let U 5,4 g+2,n+2 denote the set of U 5 g+2,n+2 -elements in which all H γi , i = 1, 2, 3 are closed, i.e. U 5 g+2,n+2 = ∪4 i=0 U 5,i g+2,n+2 .…”

Section: The Main Theoremmentioning

confidence: 99%

“…For any u g+2,n+2 ∈ U 5,4 g+2,n+2 , each of the H γi is closed. Thus the restrictions α| Hγ i , for i = 1, 2, 3 are welldefined and fixed-point free involutions.…”

Section: The Main Theoremmentioning

confidence: 99%

A Bijection for Tricellular Maps

Han

Reidys

2013

ISRN Discrete Mathematics

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We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion.

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A bijection between unicellular and bicellular maps

Cited by 2 publications

References 10 publications

Enumeration of Meanders and Masur–veech Volumes

Enumeration of Meanders and Masur–veech Volumes

A Bijection for Tricellular Maps

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