Cluster algebras, snake graphs and continued fractions Ralf - PowerPoint PPT Presentation

Expansion formula for cluster variables Theorem (Musiker-S.-Williams) If γ is an arc in a triangulated surface ( S , M ) , the cluster variable x γ is given by the formula 1 � x γ = x ( P ) . cross( G γ ) P ∈ Match G γ b 1 2 x 1 a 2 d c γ 2 1 b 1 2 c 1 2 x 3 3 3 1 a 1 2 d f 1 2 x 2 G γ e x γ = x 1 + x 2 + x 3 x 1 x 2

Example 2 1 2 2 1 2 a b 1 2 1 a b a 2 1 2 G γ γ b x 2 1 2 1 1 2 1 2 1 1 1 1 1 2 1 1 2 1 x 2 x 2 2 2 1 1 1 2 1 x 4 2 x γ = x 2 1 +1+2 x 2 2 + x 4 2 x 2 1 x 2

Example 3 1 3 2 3 1 3 2 3 3 2 2 1 3 2 3 1 [2 , 2 , 2 , 2 , 2 , 2 , 2 , 2] = 985 1 3 2 3 408 Torus with 1 puncture The Laurent polynomial of this cluster variable has 985 terms.

Expansion formula as continued fraction Theorem (C ¸anak¸ cı-S) The cluster variable x γ of the arc γ is the numerator of the continued fraction [ L 1 , L 2 , . . . , L n ] , where L i is a Laurent polynomial explicitly given 3 by the subgraph H i . 3 [C ¸anak¸ cı-Felikson]

Example 3 1 3 2 3 1 3 2 3 3 2 2 1 3 2 3 1 [2 , 2 , 2 , 2 , 2 , 2 , 2 , 2] = 985 1 3 2 3 408 Torus with 1 puncture 3 = x 2 1 + x 2 [ a 1 , . . . , a 8 ] = [2 , 2 , 2 , 2 , 2 , 2 , 2 , 2], let x ′ , then 2 x 3 x 2 x 7 1 x 2 L 1 = x ′ , L 2 = x ′ , L 3 = x ′ 1 , . . . , L 8 = x ′ 2 3 3 3 3 x 2 x 3 x 8 x 2 1 2 1 x γ = numerator of [ L 1 , L 2 , . . . , L n ]

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What about the denominator? 1 3 2 3 1 3 2 3 3 2 2 1 3 2 3 1 [2 , 2 , 2 , 2 , 2 , 2 , 2 , 2] = 985 1 3 2 3 408 Torus with 1 puncture x γ = numerator of [ L 1 , L 2 , . . . , L n ]

What about the denominator? 1 3 2 3 1 3 2 3 3 2 2 1 3 2 3 1 1 3 Torus with 1 puncture x γ ′ = denominator of [ L 1 , L 2 , . . . , L n ] = numerator of [ L 2 , . . . , L n ]

Asymptotic behavior of quotients � 1 − x 1 ) 2 + 4 x ′ 2 x ′ ( x ′ 1 − x 1 + x γ 3 lim x γ ′ = . 2 x ′ n →∞ 3 where 3 = ( x 2 1 + x 2 1 = ( x 2 2 + x ′ 2 x ′ x ′ 2 ) / x 3 3 ) / x 1 are obtained from the initial cluster by mutation in 3 and then 1.

Applications to elementary Number Theory ◮ (skein) relations in the cluster algebras were expressed in terms of snake graphs [C ¸anak¸ cı-S.] (snake graph calculus). ◮ This provides a long list of relations in terms of snake graphs. ◮ Translating into the language of continued fractions gives a long list of relations there.

Equations for continued fractions Theorem We have the following identities of numerators of continued fractions, where we set N [ a 1 , . . . , a 0 ] = 1 , and N [ a n +1 , . . . , a n ] = 1 . (a) For every i = 1 , 2 , . . . , n, N [ a 1 , . . . , a n ] N [ a 1 , . . . , a i ] N [ a i +1 , . . . , a n ] = N [ a 1 , . . . , a i − 1 ] N [ a i +2 , . . . , a n ] . + (b) For every j ≥ 0 and i such that 1 ≤ i + j ≤ n − 1 , N [ a 1 , . . . , a i + j ] N [ a i , . . . , a n ] = N [ a 1 , . . . , a n ] N [ a i , . . . , a i + j ] ( − 1) j N [ a 1 , . . . , a i − 2 ] N [ a i + j +2 , . . . , a n ] . +

Equations for continued fractions (c) For continued fractions [ a 1 , . . . , a n ] and [ b 1 , . . . , b m ] such that [ a i , . . . , a i + k ] = [ b j , . . . , b j + k ] for certain i , j , k , we have N [ a 1 , . . . , a n ] N [ b 1 , . . . , b m ] = N [ a 1 , . . . , a i − 1 , b j , . . . , b m ] N [ b 1 , . . . , b j − 1 , a i , . . . , a n ] +( − 1) k N [ a 1 , . . . , a i − 2 − 1 , 1 , b j − 1 − a i − 1 − 1 , b j − 2 , . . . , b 1 ] N ′ where N ′ =  N [ b m , . . . , b j + 2 − 1 , 1 , a i + 1 − b j + 1 − 1 , a i + 2 , . . . , a n ] k + k + k + k +   if a i + k +1 > b j + k +1 ;    N [ b m , . . . , b j + 1 − a i + 1 − 1 , 1 , a i + 2 − 1 , , . . . , a n ] 2 , b j +  k + k + k + k +    if a i + k +1 < b j + k +1 . 

180 ◦ Rotation

180 ◦ Rotation

180 ◦ Rotation

180 ◦ Rotation

180 ◦ Rotation

180 ◦ Rotation

180 ◦ Rotation

� 180 ◦ Rotation rotation reverse � [3 , 2 , 1 , 3 , 2] [2 , 3 , 1 , 2 , 3] Theorem The numerators of the continued fractions [ a 1 , a 2 , . . . , a n ] and [ a n , . . . , a 2 , a 1 ] are equal.

Palindromic snake graphs A continued fraction [ a 1 , . . . , a n ] is called ◮ even if n is even;

Palindromic snake graphs A continued fraction [ a 1 , . . . , a n ] is called ◮ even if n is even; ◮ palindromic if a i = a n − i for all i .

Palindromic snake graphs A continued fraction [ a 1 , . . . , a n ] is called ◮ even if n is even; ◮ palindromic if a i = a n − i for all i . . A snake graph is called ◮ palindromic if it is the snake graph of an even palindromic continued fraction;

Palindromic snake graphs A continued fraction [ a 1 , . . . , a n ] is called ◮ even if n is even; ◮ palindromic if a i = a n − i for all i . . A snake graph is called ◮ palindromic if it is the snake graph of an even palindromic continued fraction; ◮ rotationally symmetric at a center tile if the rotation about 180 ◦ at the center of the central tile is an automorphism.

Palindromic snake graphs A continued fraction [ a 1 , . . . , a n ] is called ◮ even if n is even; ◮ palindromic if a i = a n − i for all i . . A snake graph is called ◮ palindromic if it is the snake graph of an even palindromic continued fraction; ◮ rotationally symmetric at a center tile if the rotation about 180 ◦ at the center of the central tile is an automorphism. Theorem A snake graph is palindromic if and only if it has a rotational symmetry at its center tile.

Example [2 , 2 , 2 , 2] = 29 12

Example N [2 , 2 , 2 , 2] = 29

Example N [2 , 2 , 2 , 2] N [2 , 2] N [2 , 2] + N [2] N [2] = 29 = 5 ∗ 5 + 2 ∗ 2 Theorem (Palindromification) Let [ a 1 , a 2 , . . . , a n ] = p n . Then q n p 2 n + q 2 n [ a n , . . . , a 2 , a 1 , a 1 , a 2 , . . . , a n ] = . p n − 1 p n + q n − 1 q n

Example N [2 , 2 , 2 , 2] N [2 , 2] N [2 , 2] + N [2] N [2] = 29 = 5 ∗ 5 + 2 ∗ 2 Theorem (PalindromificationnoitacifimordnilaP) Let [ a 1 , a 2 , . . . , a n ] = p n . Then q n p 2 n + q 2 n [ a n , . . . , a 2 , a 1 , a 1 , a 2 , . . . , a n ] = . p n − 1 p n + q n − 1 q n

Sums of two squares An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd( p , q ) = 1 such that N = p 2 + q 2 .

Sums of two squares An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd( p , q ) = 1 such that N = p 2 + q 2 . Corollary Let N > 0 . ◮ If N is a sum of two squares then there exists a palindromic snake graph G such that m ( G ) = N.

Sums of two squares An integer N is called a sum of two squares if there exist integers p > q ≥ 1 with gcd( p , q ) = 1 such that N = p 2 + q 2 . Corollary Let N > 0 . ◮ If N is a sum of two squares then there exists a palindromic snake graph G such that m ( G ) = N. ◮ The number of ways one can write N as a sum of two squares is equal to one half of the number of palindromic snake graphs with N perfect matchings. Example 5 can be written uniquely as sum of two squares as 5 = 2 2 + 1 2 . The even palindromic continued fractions with numerator 5 are [2 , 2] and [1 , 1 , 1 , 1].

Markov numbers A triple of positive integers ( a , b , c ) is called a Markov triple if a 2 + b 2 + c 2 = 3 abc . An integer is called a Markov number if it is a member of a Markov triple. (29 , 2 , 169) ❦ ❦ ❦ ❦ (29 , 2 , 5) (29 , 433 , 5) ❧ ❧ ❧ (1 , 1 , 1) (1 , 2 , 1) (1 , 2 , 5) ❘ ❘ ❘ (1 , 13 , 5) ❙ (194 , 13 , 5) ❙ ❙ ❙ Markov tree (1 , 13 , 34) Uniqueness Conjecture (Frobenius 1913) The largest integer in a Markov triple determines the other two.

Markov numbers as numbers of perfect matchings of Markov snake graphs ◮ Markov triples are related to the clusters of the cluster algebra of the torus with one puncture [Beineke-Br¨ ustle-Hille 11, Propp –]. ◮ A Markov snake graph is the snake graph of a cluster variable of the once punctured torus. ◮ The Markov numbers are precisely the number of perfect matchings of the Markov snake graphs. [Propp] Uniqueness Conjecture Any two Markov snake graphs have a different number of perfect matchings.

Markov tree — snake graph version

Markov snake graphs (7,3) (7,3) 3 3 2 2 1 1 0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 The line with slope p / q = 3 / 7 with its lower Christoffel path in red, defining the Christoffel word x , x , x , y , x , x , y , x , x , y . The corresponding Markov snake graph is obtained by placing tiles of side length 1/2 on the Christoffel path leaving the first half step and the last half step empty.

Theorem Every Markov number is the numerator of an even palindromic continued fraction. Corollary Every Markov number, except 1, is a sum of two squares. In general, the decomposition of an integer as a sum of two squares is not unique.The smallest example among the Markov numbers is 610 = 23 2 + 9 2 = 21 2 + 13 2 . 21 / 13 = [1 , 1 , 1 , 1 , 1 , 2] and its palindromification [2 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2] is a Markov snake graph. 23 / 9 = [2 , 1 , 1 , 4] and its palindromification [4 , 1 , 1 , 2 , 2 , 1 , 1 , 4] is not Markov.

Chebyshev polynomials The (normalized) Chebyshev polynomials of the first kind T n are defined recursively by T 0 = 1 , T 1 = x , and T n = xT n − 1 − T n − 2 . The first few polynomials are x 2 − 1 = T 2 x 3 − 2 x T 3 = x 4 − 3 x 2 + 1 = T 4 x 5 − 4 x 3 + 3 x T 5 = x 6 − 5 x 4 + 6 x 2 − 1 T 6 =

Chebyshev polynomials Let G n be the snake graph of [ a 1 , a 2 , . . . , a n ] = [1 , 1 , . . . , 1]. Thus G n is a vertical straight snake graph with exactly n − 1 tiles. 5 4 3 G 6 2 1