Drawing Planar Graphs via Dessins d’Enfants Kevin Bowman, Sheena Chandra, Anji Li and Amanda Llewellyn PRiME 2013: Purdue Research in Mathematics Experience, Purdue University July 21, 2013 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 1 / 25
Belyi Maps Definition A rational function β ( z ) = p ( z ) q ( z ) , where β : P 1 ( C ) → P 1 ( C ), is a Belyi Map if it has at most three critical values , say { ω (0) , ω (1) , ω ( ∞ ) } . Remarks : P 1 ( C ) refers to the complex projective line, that is the set C ∪ {∞} . An ω ∈ P 1 ( C ) is a critical value of β ( z ) if β ( z ) = ω for some β ′ ( z ) = 0. Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 2 / 25
Belyi Maps Question Is β ( z ) = 4 z 5 (1 − z 5 ) a Belyi Map? 1 Form a polynomial equation β ( z ) = ω 1 ω 0 4 z 5 (1 − z 5 ) − ω 1 = 0 ⇐ ⇒ ω 0 2 Compute the discriminant � � ω 0 4 z 5 (1 − z 5 ) − ω 1 = ( constant ) ω 4 1 ω 9 0 ( ω 1 − ω 0 ) 5 disc 3 Find the roots � ω 1 � � � 0 ( ω 1 − ω 0 ) 5 = 0 ∈ P 1 ( C ) � ω 4 1 ω 9 = = { 0 , 1 , ∞} � ω 0 Answer Yes, β ( z ) is a Belyi Map with critical values { 0 , 1 , ∞} Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 3 / 25
Dessin d’Enfant Definition Given a Belyi map β ( z ) = p ( z ) / q ( z ) we consider the preimages = β − 1 (0) B = “black” vertices = β − 1 (1) W = “white” vertices = β − 1 ([0 , 1]) E = edges = β − 1 ( ∞ ) F = midpoints of faces We define the bipartite graph ∆ β = ( B ∪ W , E ) as the Dessin d’Enfant . Remarks: A bipartite graph is a collection of vertices and edges where the vertices are placed into two disjoint sets, none of whose elements are adjacent Following Grothendieck, “Dessin d’Enfants” is French for “Children’s Drawings” Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 4 / 25
Dessin d’Enfant Example β ( z ) = 4 z 5 (1 − z 5 ) We found that β is a Belyi map with critical values 0 , 1 , ∞ . We consider its preimages B = β − 1 (0) = { 0 } ∪ { 5 th roots of 1 } W = β − 1 (1) = { 5 th roots of 1 / 2 } F = β − 1 ( ∞ ) = {∞} Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 5 / 25
Research Question Motivating Question Let Γ be a connected planar graph. Can we find a Belyi map β ( z ) such that Γ is the Dessin d’Enfant of this map? Remarks: A graph is connected if there is a path between any two points A graph is planar if it can be drawn without any edges crossing Any planar graph can be seen as a bipartite graph if we label all vertices as “black” and label all midpoints of edges as “white” Specific Question Given a specific web or a tree, can we explicitly find its corresponding Belyi map? Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 6 / 25
Methodology From a graph to a bipartite graph: 1) Label graph’s vertices as “black” 2) Add “white” vertices as midpoints of edges Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 7 / 25
Methodology From a bipartite graph to a Belyi map: 3) Label ”black” vertices as B = { b 1 , b 2 , ..., b r } ⊆ P 1 ( C ) such that each point b k has e k edges incident. Since we want B = β − 1 (0), we must have r β ( z ) = p ( z ) � ( z − b k ) e k p ( z ) = ( constant ) where q ( z ) k =1 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 8 / 25
Methodology 4) Label “white” vertices as W = { w 1 , w 2 , ..., w n } ⊆ P 1 ( C ) such that each point w k has 2 edges incident. Since we want W = β − 1 (1), we must have n β ( z ) = p ( z ) � ( z − w k ) 2 p ( z ) − q ( z ) = ( constant ) where q ( z ) k =1 5) Label midpoints of faces vertices as F = { f 1 , f 2 , ..., f s } ⊆ P 1 ( C ) such that each point f k has d k edges that enclose it. Since we want F = β − 1 ( ∞ ), we must have s � ( z − f k ) d k q ( z ) = ( constant ) k =1 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 9 / 25
Methodology Proposition If there exist constants b k , w k , f k , p 0 , q 0 , r 0 ∈ C such that r n s � � � ( z − w k ) 2 ( z − b k ) e k ( z − f k ) d k p 0 − q 0 − r 0 = 0 k =1 k =1 k =1 � �� � � �� � � �� � vertices edges faces for all z , then the rational function � r k =1 ( z − b k ) e k β ( z ) = − p 0 � s k =1 ( z − f k ) d k r 0 is a Belyi map of degree r s � � 2 n = e k = d k . k =1 k =1 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 10 / 25
List of Results Cycles Dipoles Prisms Bipyramids Webs Antiprisms Trapezohedrons Wheels Gyroelongated Bipyramid Truncated Trapezohedron Paths Trees Stars Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 11 / 25
Graphing: Mathematica Notebook http://www.math.purdue.edu/ ∼ egoins/site//Dessins%20d’Enfants.html Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 12 / 25
Cycles β ( z ) = − ( z n − 1) 2 4 z n Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 13 / 25
Dipoles 4 z n β ( z ) = − ( z n − 1) 2 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 14 / 25
Prisms β ( z ) = ( z 2 n + 14 z n + 1) 3 108 z n ( z n − 1) 4 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 15 / 25
Bipyramids 108 z n ( z n − 1) 4 β ( z ) = ( z 2 n + 14 z n + 1) 3 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 16 / 25
Antiprisms ( z 2 n + 10 z n − 2) 4 β ( z ) = 16( z n − 1) 3 (2 z n + 1) 3 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 17 / 25
Trapezohedron β ( z ) = 16( z n − 1) 3 (2 z n + 1) 3 ( z 2 n + 10 z n − 2) 4 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 18 / 25
Wheels β ( z ) = − 64 z n ( z n − 1) 3 (8 z n + 1) 3 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 19 / 25
Gyroelongated Bipyramid 1728 z n ( z 2 n + 11 z n − 1) 5 β ( z ) = − ( z 4 n − 228 z 3 n + 494 z 2 n + 228 z n + 1) 3 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 20 / 25
Truncated Trapezohedron β ( z ) = − ( z 4 n − 228 z 3 n + 494 z 2 n + 228 z n + 1) 3 1728 z n ( z 2 n + 11 z n − 1) 5 Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 21 / 25
Paths β ( z ) = sin 2 ( n cos − 1 z ) Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 22 / 25
Stars β ( z ) = 4 z n (1 − z n ) Wikipedia Mathematica Notebook Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 23 / 25
Further Research Find the Belyi maps for the following: Elongated Pyramid Gyroelongated Pyramid Rotunda Elongated Bipyramid Truncated Bipyramid Bicupola Birotunda Create a Mathematica notebook which will generate Belyi maps for any given tree or web. Find a Belyi map for the Stick Figure P 1 P 3 P 2 P 4 P 6 P 5 P 7 Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 24 / 25
Acknowledgements The National Science Foundation (NSF) and Professor Steve Bell Dr. Joel Spira (Purdue BS ’48, Physics) Andris “Andy” Zoltners (Purdue MS ’69, Mathematics) Professor Edray Goins Andres Figuerola Drawing Planar Graphs via Dessins d’Enfants July 21, 2013 25 / 25
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