visualizing dessins d’enfants - willamette...

IntroductionFinding Belyi Maps

The Pell-Abel Equation: An ApplicationClosing Remarks

Visualizing Dessins D’EnfantsWillamette Valley Consortium for Mathematics Research

Mary Kemp and Susan Maslak

Occidental College and Ave Maria University

MAA MathFestAugust 7, 2014

Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants



IntroductionBelyi MapsDessinsPassportsShabat Polynomials

Finding Belyi MapsQuestionExampleGeneral Results

The Pell-Abel Equation: An Application

Closing Remarks




Belyi Maps

DefinitionA Belyi Map is a function F : X → C with the critical values inthe set {0, 1,∞}

I X is a Riemann SurfaceI C = C ∪ {∞} is equivalent to a sphere in R3




DefinitionA Dessin D’Enfant, or Dessin for short, is a connected bicoloredgraph that is embedded on a Riemann surface and is obtained by aBelyi function, f , in the following way:

I F−1(0)→ black vertices

I F−1(1)→ white vertices (shown as red in our pictures)

I F−1([0, 1])→ edges




Examples of Dessins and Their Belyi Maps

0.1 0.2 0.3 0.4 0.5 0.6Re�z�

�0.4

�0.2

0.2

0.4

Im�z� z � �3125 �4 z � 1�6186 624 z

�1.0 �0.5 0.5 1.0Re�z�

�1.5

�1.0

�0.5

0.5

1.0

1.5Im�z�z �

�1� z�3 �35 z4 � 40 z3 � 48 z2 � 64 z � 128�64 398046511104

−3125(4z−1)6

186624z(1−z)3(35z4+40z3+48z2+64z+128)6

4398046511104




Passports

DefinitionA Passport is a set of valencies (or degrees) that corresponds tothe vertices of the dessin, which is represented in the form[b1, b2, ..., bn; w1,w2, ...,wk ] where b1....bn are the the blackdegrees and w1, ...,wk are the white degrees.

Example

[5, 4, 3; 3, 1, 1, 1, 1, 1, 1, 1, 1, 1] can also be written as [5, 4, 3; 3, 19]




Shabat Polynomials

I A Shabat Polynomial refers to a Belyi Map defined byF : C 7→ C, which is represented by a tree (a graph with nocycles).

I given a tree we can find its Shabat polynomial by solving:I F (z) = 0 → black verticesI F (z)− 1 = 0 → white vertices




Finding Belyi Maps

QuestionCan we find all Belyi maps for a given passport of size k?

[33, 15; 27]→

-1.0 -0.5 0.5 1.0ReHzL

-2

-1

1

2

ImHzLz ®

4 H1 - zL z3 I2 z

2 + 3 z + 9M3 I8 z4 + 28 z

3 + 126 z2 + 189 z + 378M

531 441




Example

The passport [s2, r 2; 4, 1(r−1)2+(s−1)2] is size k = 2.

[42, 22; 4, 18]




F (z) = A(z − c0)4(z − c1)4(z − c2)2(z − c3)2

F (z)− 1 = A(z − d0)4(z8 + d8z7 + d7z

6 + · · ·+ d2z + d1)

[42, 22; 4, 18]

Belyi map when s = 4, r = 2:

F (z) = 1256 (2z2 + 4)4(z2 − 1)2

General Form:F (z) = s−s(−1)r (z2−1)r (s + rz2)s




Belyi Maps for Passports with only one Dessin (k=1 inShabat-Zvonkin paper)

Passport Belyi Map[n; 1n] zn

[2n, 1; 2n, 1]1 + cos ((2n + 1) arccos(z))

2

[2n; 2n−1, 12]1 + cos (2n arccos(z))

2

[sr−1, t; r , 1(r−1)(s−1)+(t−1)] (1− z)t

(r−1∑k=0

( t

s

)k

zk

k!

)s

[r , t, 1r+t−2; 2r+t−1] 4Sr ,t(z)(1− Sr ,t(z))

[n2, 14n−3; 32n−1] −3√

3 i Sr (z) (1− Sr (z))(

Sr (z)− 1−i√

32

)[33, 15; 27] − 4

531441 (z − 1)z3(2z2 + 3z + 9

)3 (8z4 + 28z3 + 126z2 + 189z + 378

)Where Sr,t (z) = (1 − z)t

r−1∑j=0

(t − 1 + j

t − 1

)z j and Sr (z) = Sr,r (z)




Belyi Maps for passports with exactly two Dessins (k=2)

Passport Belyi Map

[r 2, s2; 4, 12r+2s−4]s−s(−1)r (z2 − 1)r (s + rz2)s

?

[r , t, 12r+2t−3); 3r+t−1]−3√

3 i Sr ,t(z) (1− Sr ,t(z))(

Sr ,t(z)− 1−i√

32

)?

[r 2, 12+3k ; 4k ]−16(−1 + Sr (z))Sr (z)(1/2− Sr (z) + Sr (z)2)

4(−1 + Sr (z))Sr (z)((−1 + i)− (1 + 2i)Sr (z) + Sr (z)2)

[r 2, 13+4k ; 5k ]

C (−1 + Sr (z))Sr (z)[i(−2 +√

5) +√

5 + 2√

5)− (−3i + 2i√

5 +√5(5− 2

√5))Sr (z) + (3i +

√5(5− 2

√5))Sr (z)2 − 2iSr (z)3]

C (−1 + Sr (z))Sr (z)(i(2 +√

5 + i√

5 + 2√

5)− (3i + 2i√

5 +√5(5 + 2

√5))Sr (z) + (−3i +

√5(5 + 2

√5))Sr (z)2 + 2iSr (z)3)

[32, 1; 22, 13]C (11i +

√7− 8iz)(−1 + z)3z3

C (3i +√

7 + 8iz)(−1 + z)3z3

[3, 22; 22, 13]Cz3(35(5+3

√21−

√70 + 30

√21)−21(17+3

√21−

√70 + 30

√21)z +180z2)2

Cz3(−35(−5 + 3√

21 + i√−70 + 30

√21) + 21(−17 + 3

√21 +

i√−70 + 30

√21)z + 180z2)2

[3, 22, 13; 25]−1

1024 (z − 1)z3(8 + 4z + 3z2)2(40 + 15z + 9z2)−4729 (z − 1)2z2(3 + 2z)3(−45− 10z + 20z2 + 8z3)

[43, 18; 210]−4(−2+z)4(−1+z)4z4(5−8z +4z2)(−1−4z−10z2−20z3+45z4−24z5+4z6)

−125389989167104 z4(36− 6z + z2)4(6 + 4z + z2)(116640− 46656z + 12960z2 −

2160z3 + 270z4 − 20z5 + z6)

Where C is a large constant (too large for the table!)




An Application

Let F (z) = 1256 (2z2 + 4)4(z2 − 1)2 be our equation found earlier.

-1.0 -0.5 0.5 1.0ReHzL

-1.5

-1.0

-0.5

0.5

1.0

1.5

ImHzL

z ®1

256Iz2 - 1M2 I2 z

2 + 4M4

If D(z) =∏

(z − di ) = (3 + z2)(−8 + 3z4 + z6)Then P(z) = 1

8 (−8 + 24z4 + 8z6 − 9z8 − 6z10 − z12)and Q(z) = 1

8 z2(−1 + z2)(2 + z2)2

is a solution to the equation P(z)2 − D(z)Q(z)2 = 1Mary Kemp and Susan Maslak Visualizing Dessins D’Enfants



The Pell-Abel Equation

QuestionSolve for P(z)2 − D(z)Q(z)2 = 1 for a given D(z)

TheoremLet T (z) be a Shabat Polynomial. Then there exists acorresponding D(z) of the form D(z) =

∏(z − di ), where di

represent the coordinates of the odd vertices, for whichP(z)2 − D(z)Q(z)2 = 1 has infinitely many solutions.

ExampleR(z) = s−s(−1)r (z2 − 1)r (s + rz2)s where s = 2n and r = 2

D(z) =

(n∑

k=1

(n+1k+1

)knn−k−1z2k−2

)(−2nn +

n∑k=1

(n+1k+1

)knn−k−1z2k+2

)where −2R(z) + 1 and [−2R(z)+1]2−1

D(z) is one of infinitely many solutions.




Future Projects and Open Questions

I Inverse Enumeration for Belyi Maps with multiple faces

I Belyi Maps for Cartesian Products of graphs

I Exploration of the role Galois Action plays on Belyi Maps andDessins

I Other applications Belyi Maps have on Diophantine Equations




Acknowledgements

I Willamette Valley Consortium for Mathematics Research, NSF

I Dr. Naiomi Cameron

I Lewis and Clark College

I Ave Maria University and Occidental College

I Dr. Edray Goins from Purdue University

I Computer Science Whizzes




References

J. Couveignes, Calcul et rationalite de fonctions de Belyi en genre 0, Annales de l’Institut

Fourier (Grenoble) 44 (1994), no. 1, 1–38.

L. Granboulan, Calcul d’objets geometriques a l’aide de methodes algebraiques et

numeriques: dessins d’enfants, Ph.D. thesis, Universite Paris 7, 1997.

Y. Kochetkov, Geometry of planar trees. (Russian) Fundam. Prikl. Mat. 13 (2007), no. 6,

149–158; translation in J. Math. Sci. (N. Y.) 158 (2009), no. 1, 106113.

S. Lando, A. Zvonkin, Graphs on surfaces and their applications, Encyclopedia of

Mathematical Sciences, Low-Dimensional Topology, II, Springer-Verlag, Berlin, 2004.

Y. Matiyasevich, Computer evaluation of generalized Chebyshev polynomials, em Moscow

Univ. Math. Bull. 51 (1996), no. 6, 39–40.

G. Shabat, A. Zvonkin, Plane trees and algebraic numbers, (English summary) Jerusalem

combinatorics ’93, 233275, Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994.

L. Schneps, The Grothendieck theory of dessins d’enfants (Luminy, 1993), London Math.

Soc. Lecture Note Ser., 200, Cambridge Univ. Press, Cambridge, 1994.

J. Sijsling, J. Voight, On computing Belyi maps, arXiv: 1311.2529v3 [math.NT], 2014.




THANK YOU!ANY QUESTIONS?


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