From Christoffel words to braids

Transcription

1 From Christoffel words to braids Christian Kassel Institut de Recherche Mathématique Avancée CNRS - Université Louis Pasteur Strasbourg, France Colloquium at Rutgers 13 April 2007

2 Joint work With Christophe Reutenauer, UQAM, Montréal, Canada Paper published in Annali di Matematica Pura ed Applicata 186 (2007), Downloadable from arxiv:math.gr/

3 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

4 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

5 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

6 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

7 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

8 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

9 What is it about? Construct words from simple planar geometry Words on two letters Christoffel words: finite words Sturmian sequences: infinite words In relation to bases of the free group F2 on two generators the braid group B4 on four strands

10 I. HISTORICAL BACKGROUND

11 Elwin Bruno Christoffel ( ) Worked on conformal mappings, geometry and tensor analysis (Christoffel symbols), theory of invariants, orthogonal polynomials, continued fractions, and applications to the theory of shock waves, to the dispersion of light. Held positions at Polytechnicum Zurich, TU Berlin, and... in Strasbourg After French-Prussian War in 1870, France lost Alsace-Lorraine to the German Empire The Prussians created a new university in Strasbourg Christoffel founded the Mathematisches Institut in 1872

12 Elwin Bruno Christoffel ( ) Worked on conformal mappings, geometry and tensor analysis (Christoffel symbols), theory of invariants, orthogonal polynomials, continued fractions, and applications to the theory of shock waves, to the dispersion of light. Held positions at Polytechnicum Zurich, TU Berlin, and... in Strasbourg After French-Prussian War in 1870, France lost Alsace-Lorraine to the German Empire The Prussians created a new university in Strasbourg Christoffel founded the Mathematisches Institut in 1872

13 Elwin Bruno Christoffel ( ) Worked on conformal mappings, geometry and tensor analysis (Christoffel symbols), theory of invariants, orthogonal polynomials, continued fractions, and applications to the theory of shock waves, to the dispersion of light. Held positions at Polytechnicum Zurich, TU Berlin, and... in Strasbourg After French-Prussian War in 1870, France lost Alsace-Lorraine to the German Empire The Prussians created a new university in Strasbourg Christoffel founded the Mathematisches Institut in 1872

14 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

15 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

16 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

17 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

18 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

19 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

20 Christoffel s Observatio arithmetica Observatio arithmetica, Annali di Matematica Pura ed Applicata, vol. 6 (1875), Published in Latin (he also published papers in German, French, Italian) Author s name appears on paper as auctore E. B. Christoffel, prof. Argentinensi Argentinensi derived from Argentoratum, name of the city founded by the Romans, later Strasbourg

21 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

22 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

23 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

24 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

25 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

26 Observatio arithmetica 1 Designantibus a, b numeros positivos integros et primos inter se, sint r 1, r 2, r 3,... residua minima non negativa numerorum a, 2a, 3a,... secundum modulum b,... Denoting coprime positive integers a, b, let r 1, r 2, r 3,... be the minimal non-negative remainders of the numbers a, 2a, 3a,... modulo b,... Agitur autem de quaestione, quando r m crescat vel decrescat, si m unitate augitur... The question is whether r m increases or decreases if m grows by one...

27 Observatio arithmetica 2... notatur littera c vel d, prout r m crescit vel decrescit. Hoc modo nova series nascitur, e duabus tantum literis c, d, sed certo quodam ordine composita write down the letter c or d according as r m increases or decreases. In this way a new sequence arises, made of the two letters c, d arranged in a certain precise order...

28 Observatio arithmetica 2... notatur littera c vel d, prout r m crescit vel decrescit. Hoc modo nova series nascitur, e duabus tantum literis c, d, sed certo quodam ordine composita write down the letter c or d according as r m increases or decreases. In this way a new sequence arises, made of the two letters c, d arranged in a certain precise order...

29 Observatio arithmetica 2... notatur littera c vel d, prout r m crescit vel decrescit. Hoc modo nova series nascitur, e duabus tantum literis c, d, sed certo quodam ordine composita write down the letter c or d according as r m increases or decreases. In this way a new sequence arises, made of the two letters c, d arranged in a certain precise order...

30 Observatio arithmetica 3 Exemplum I. Sit a = 4, b = 11, erit series (r. ) notis c, d ornata r. = g. = c d c c d c c d c d c The integers a and b can be recovered from the sequence (g. ): a is the number of d s in (g. ) b is the length of (g. ) Main result of the paper tells how to recover the continued fraction expansion of a/b from (g. )

31 Observatio arithmetica 3 Exemplum I. Sit a = 4, b = 11, erit series (r. ) notis c, d ornata r. = g. = c d c c d c c d c d c The integers a and b can be recovered from the sequence (g. ): a is the number of d s in (g. ) b is the length of (g. ) Main result of the paper tells how to recover the continued fraction expansion of a/b from (g. )

32 Observatio arithmetica 3 Exemplum I. Sit a = 4, b = 11, erit series (r. ) notis c, d ornata r. = g. = c d c c d c c d c d c The integers a and b can be recovered from the sequence (g. ): a is the number of d s in (g. ) b is the length of (g. ) Main result of the paper tells how to recover the continued fraction expansion of a/b from (g. )

33 Observatio arithmetica 3 Exemplum I. Sit a = 4, b = 11, erit series (r. ) notis c, d ornata r. = g. = c d c c d c c d c d c The integers a and b can be recovered from the sequence (g. ): a is the number of d s in (g. ) b is the length of (g. ) Main result of the paper tells how to recover the continued fraction expansion of a/b from (g. )

34 Observatio arithmetica 3 Exemplum I. Sit a = 4, b = 11, erit series (r. ) notis c, d ornata r. = g. = c d c c d c c d c d c The integers a and b can be recovered from the sequence (g. ): a is the number of d s in (g. ) b is the length of (g. ) Main result of the paper tells how to recover the continued fraction expansion of a/b from (g. )

35 Observatio arithmetica 4 Sequences of letters c et d thus obtained, e.g., are called Christoffel words cdccdccdcdc,

36 Christoffel words (and infinite variants) come up in mathematics symbolic dynamics (Morse) continued fractions computer science formal language theory algorithms on words pattern recognition physics biology crystallography

37 II. A GEOMETRICAL CONSTRUCTION OF CHRISTOFFEL WORDS

38 Primitive vectors of Z 2 and Christoffel words ( p q) Z 2 is primitive if p, q are coprime To ( p q) we shall attach a Christoffel word in the letters a, a 1, b, b 1 w = w ( ) p q w represents an element of free group F2 = F (a, b) the image of w in Z 2 is ( ) p q

39 Primitive vectors of Z 2 and Christoffel words ( p q) Z 2 is primitive if p, q are coprime To ( p q) we shall attach a Christoffel word in the letters a, a 1, b, b 1 w = w ( ) p q w represents an element of free group F2 = F (a, b) the image of w in Z 2 is ( ) p q

40 Primitive vectors of Z 2 and Christoffel words ( p q) Z 2 is primitive if p, q are coprime To ( p q) we shall attach a Christoffel word in the letters a, a 1, b, b 1 w = w ( ) p q w represents an element of free group F2 = F (a, b) the image of w in Z 2 is ( ) p q

41 Primitive vectors of Z 2 and Christoffel words ( p q) Z 2 is primitive if p, q are coprime To ( p q) we shall attach a Christoffel word in the letters a, a 1, b, b 1 w = w ( ) p q w represents an element of free group F2 = F (a, b) the image of w in Z 2 is ( ) p q

42 Planar representation of a primitive vector If P = ( p q) is primitive, then [OP] Z 2 = {O, P} O = (0, 0) P = ( ) p q

43 Stair-case approximation If p, q 0, then approximate segment OP by closest stair-case path from beneath P = ( ) p q O

44 Christoffel word of a positive primitive vector To the vector ( 5 3) N 2 we attach the word w ( 5 3) = aabaabab = a 2 ba 2 bab F 2 ( 5 3) a b O a a b a b a

45 Christoffel words: factorization property 1 O : closest point of Z 2 to segment OP w ( ) 5 3 = (a 2 b)(a 2 bab) = w ( ( 2 1) w 3 ) 2 b a a b a a b a P = ( ) 5 3

46 Christoffel words: factorization property 2 Theorem If ( p q), ( r s) N 2 satisfy w p q r s = 1, then ( ) p + r = w q + s ( ) p w q ( ) r s

47 General Christoffel words ā = a 1 b = b 1 ba 1 ba 1 ba 2 ( 4 ) ( ā ā 5 3 3) b b ā a a 2 ba 2 bab b b ā a a b b a a a b b O ( a a 3 ) a 2 3 b 1 a 2 b 1 b ā b ( 5 ) 2 ā ā b 1 a 1 b 1 a 2

48 Bases of the free group F 2 Let F 2 = F(a, b) be the free group on two generators a, b {u, v} F 2 is a basis if {u, v} generates F 2 Equivalently, {u, v} is a basis if there is ϕ Aut(F 2 ) with ϕ(a) = u and ϕ(b) = v Two bases {u, v} and {u, v } of F 2 are conjugate if there is w F 2 such that u = wuw 1 and v = wvw 1

49 Bases of the free group F 2 Let F 2 = F(a, b) be the free group on two generators a, b {u, v} F 2 is a basis if {u, v} generates F 2 Equivalently, {u, v} is a basis if there is ϕ Aut(F 2 ) with ϕ(a) = u and ϕ(b) = v Two bases {u, v} and {u, v } of F 2 are conjugate if there is w F 2 such that u = wuw 1 and v = wvw 1

50 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

51 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

52 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

53 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

54 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

55 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

56 Lifting bases of Z 2 to F 2 Recall: { ( p q), ( r s) } is a basis of Z 2 ps qr = ±1 Question: Can we lift { ( ( p q), r s) } to a basis of F2? Example: a 2 b F 2 is a lift of ( ) 2 1 a 3 b 2 is a lift of ( ) 3 {a 2 b, a 3 b 2 } is not a basis of F 2, {a 2 b, a 2 bab} is a basis of F 2 2

57 Christoffel bases of F 2 Theorem If {( p q), ( r s)} is a basis of Z 2, then { w ( ) p, w q ( )} r s {( is a basis of F 2 (called a Christoffel basis). It lifts the basis p ) ( q, r )} s to F2 Any basis of F 2 is conjugate of a (unique) Christoffel basis

58 Christoffel bases of F 2 Theorem If {( p q), ( r s)} is a basis of Z 2, then { w ( ) p, w q ( )} r s {( is a basis of F 2 (called a Christoffel basis). It lifts the basis p ) ( q, r )} s to F2 Any basis of F 2 is conjugate of a (unique) Christoffel basis

59 Christoffel bases of F 2 Theorem If {( p q), ( r s)} is a basis of Z 2, then { w ( ) p, w q ( )} r s {( is a basis of F 2 (called a Christoffel basis). It lifts the basis p ) ( q, r )} s to F2 Any basis of F 2 is conjugate of a (unique) Christoffel basis

60 Palindromes The reverse of a word w is the word w obtained by reading w from right to left A palindrome is a word w such that w = w A basis {u, v} of F 2 is palindromic if both u and v are palindromes

61 Palindromes The reverse of a word w is the word w obtained by reading w from right to left A palindrome is a word w such that w = w A basis {u, v} of F 2 is palindromic if both u and v are palindromes

62 Palindromes The reverse of a word w is the word w obtained by reading w from right to left A palindrome is a word w such that w = w A basis {u, v} of F 2 is palindromic if both u and v are palindromes

63 Existence of palindromic bases Let w be the length of a word w with respect to the alphabet {a, b, a 1, b 1 } Theorem Any basis {u, v} of F 2 with u, v odd is the conjugate of a unique (cyclically reduced) palindromic basis Example: {aba 2 b, a 2 b} is a non-palindromic basis of F 2, but is conjugated to the palindromic basis {ababa, aba}

64 III. STURMIAN SEQUENCES AND MORPHISMS

65 Infinite version of Christoffel words Let L R 2 + be a half-line originating from O and satisfying L Z 2 = {O} The slope of L is irrational The infinite sequence in a, b encoding the stair-case approximation of L is a Sturmian sequence

66 Infinite version of Christoffel words Let L R 2 + be a half-line originating from O and satisfying L Z 2 = {O} The slope of L is irrational The infinite sequence in a, b encoding the stair-case approximation of L is a Sturmian sequence

67 Sturmian sequences: formal definition Infinite word in a, b: map {0, 1, 2,...} {a, b} Example: abaababaabaab Sturmian sequence: infinite word w in a, b such that the number of distinct factors of w of length n is n + 1 for each n 1 (w 1 is a factor of w = w 0 w 1 w 2 ) n = 1: Number of distinct letters is 2 n = 2: Three possible length-two factors out of four: aa, ab, ba, bb

68 Sturmian sequences: formal definition Infinite word in a, b: map {0, 1, 2,...} {a, b} Example: abaababaabaab Sturmian sequence: infinite word w in a, b such that the number of distinct factors of w of length n is n + 1 for each n 1 (w 1 is a factor of w = w 0 w 1 w 2 ) n = 1: Number of distinct letters is 2 n = 2: Three possible length-two factors out of four: aa, ab, ba, bb

69 Sturmian sequences: formal definition Infinite word in a, b: map {0, 1, 2,...} {a, b} Example: abaababaabaab Sturmian sequence: infinite word w in a, b such that the number of distinct factors of w of length n is n + 1 for each n 1 (w 1 is a factor of w = w 0 w 1 w 2 ) n = 1: Number of distinct letters is 2 n = 2: Three possible length-two factors out of four: aa, ab, ba, bb

70 Sturmian sequences: formal definition Infinite word in a, b: map {0, 1, 2,...} {a, b} Example: abaababaabaab Sturmian sequence: infinite word w in a, b such that the number of distinct factors of w of length n is n + 1 for each n 1 (w 1 is a factor of w = w 0 w 1 w 2 ) n = 1: Number of distinct letters is 2 n = 2: Three possible length-two factors out of four: aa, ab, ba, bb

71 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

72 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

73 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

74 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

75 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

76 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

77 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

78 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

79 The Fibonacci sequence Obtain Sturmian sequences dynamically by iterating an appropriate substitution ϕ : {a, b} {a, b} Example: Let ϕ(a) = ab and ϕ(b) = a. Then w n = ϕ n (a) converges to a Sturmian sequence w w0 = a w1 = ab w2 = aba w3 = abaab w 4 = abaababa w 5 = abaababaabaab w approximates half-line of slope ( 5 1)/2 (inverse of golden ratio)

80 Sturmian morphisms A substitution replaces a et b by (positive) words in a and b (i.e., it is an endomorphism of free monoid {a, b} ) Substitutions can be composed and form a monoid A Sturmian morphism is a substitution preserving the Sturmian sequences The Sturmian morphisms form a monoid under composition, the Sturmian monoid St

81 Sturmian morphisms A substitution replaces a et b by (positive) words in a and b (i.e., it is an endomorphism of free monoid {a, b} ) Substitutions can be composed and form a monoid A Sturmian morphism is a substitution preserving the Sturmian sequences The Sturmian morphisms form a monoid under composition, the Sturmian monoid St

82 Sturmian morphisms A substitution replaces a et b by (positive) words in a and b (i.e., it is an endomorphism of free monoid {a, b} ) Substitutions can be composed and form a monoid A Sturmian morphism is a substitution preserving the Sturmian sequences The Sturmian morphisms form a monoid under composition, the Sturmian monoid St

83 Examples of Sturmian morphisms id = We have E 2 = id L a = L b = EL a E = ( ) a a b b ( ) a a b ab ( ) a ba b b and E = and R a = ( ) a b b a ( ) a a b ba and R b = ER a E = ( ) a ab b b

84 Examples of Sturmian morphisms id = We have E 2 = id L a = L b = EL a E = ( ) a a b b ( ) a a b ab ( ) a ba b b and E = and R a = ( ) a b b a ( ) a a b ba and R b = ER a E = ( ) a ab b b

85 Examples of Sturmian morphisms id = We have E 2 = id L a = L b = EL a E = ( ) a a b b ( ) a a b ab ( ) a ba b b and E = and R a = ( ) a b b a ( ) a a b ba and R b = ER a E = ( ) a ab b b

86 The Sturmian monoid as a submonoid of Aut(F 2 ) Mignosi & Séébold (1993): The monoid St is generated by {E, L a, R a } or by {E, L b, R b } Considered as endomorphisms of the free group F 2, the generators E, L a, R a of St are invertible in End(F 2 ) Therefore, St is a submonoid of the group Aut(F 2 ) St Aut(F 2 ) Note: the subgroup generated by {E, L a, R a } is Aut(F 2 )

87 The Sturmian monoid as a submonoid of Aut(F 2 ) Mignosi & Séébold (1993): The monoid St is generated by {E, L a, R a } or by {E, L b, R b } Considered as endomorphisms of the free group F 2, the generators E, L a, R a of St are invertible in End(F 2 ) Therefore, St is a submonoid of the group Aut(F 2 ) St Aut(F 2 ) Note: the subgroup generated by {E, L a, R a } is Aut(F 2 )

88 The Sturmian monoid as a submonoid of Aut(F 2 ) Mignosi & Séébold (1993): The monoid St is generated by {E, L a, R a } or by {E, L b, R b } Considered as endomorphisms of the free group F 2, the generators E, L a, R a of St are invertible in End(F 2 ) Therefore, St is a submonoid of the group Aut(F 2 ) St Aut(F 2 ) Note: the subgroup generated by {E, L a, R a } is Aut(F 2 )

89 The Sturmian monoid as a submonoid of Aut(F 2 ) Mignosi & Séébold (1993): The monoid St is generated by {E, L a, R a } or by {E, L b, R b } Considered as endomorphisms of the free group F 2, the generators E, L a, R a of St are invertible in End(F 2 ) Therefore, St is a submonoid of the group Aut(F 2 ) St Aut(F 2 ) Note: the subgroup generated by {E, L a, R a } is Aut(F 2 )

90 Projecting onto GL 2 (Z) Linearization map: π : Aut(F 2 ) Aut(Z 2 ) = GL 2 (Z) π(e) = π(l a ) = π(r a ) = A = π(l b ) = π(r b ) = B = ( The set {A, B} generates SL 2 (Z) ) ( ( ) SL 2 (Z) ) SL 2 (Z)

91 Projecting onto GL 2 (Z) Linearization map: π : Aut(F 2 ) Aut(Z 2 ) = GL 2 (Z) π(e) = π(l a ) = π(r a ) = A = π(l b ) = π(r b ) = B = ( The set {A, B} generates SL 2 (Z) ) ( ( ) SL 2 (Z) ) SL 2 (Z)

92 Projecting onto GL 2 (Z) Linearization map: π : Aut(F 2 ) Aut(Z 2 ) = GL 2 (Z) π(e) = π(l a ) = π(r a ) = A = π(l b ) = π(r b ) = B = ( The set {A, B} generates SL 2 (Z) ) ( ( ) SL 2 (Z) ) SL 2 (Z)

93 Projecting onto GL 2 (Z) Linearization map: π : Aut(F 2 ) Aut(Z 2 ) = GL 2 (Z) π(e) = π(l a ) = π(r a ) = A = π(l b ) = π(r b ) = B = ( The set {A, B} generates SL 2 (Z) ) ( ( ) SL 2 (Z) ) SL 2 (Z)

94 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

95 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

96 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

97 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

98 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

99 Presentation of SL 2 (Z) Generators: A = Defining relations: ( ) and B = Braid relation: AB 1 A = B 1 AB 1 Torsion relation: (AB 1 A) 4 = 1 ( Question: Can we lift these relations to Aut(F 2 ) using the lifts L a, R a of A and the lifts L b, R b of B? Answer: Yes, see next slide )

100 Lifting the braid relations to Aut(F 2 ) Commutation relations: [the lifts of A (or of B) commute] L a R a = R a L a and L b R b = R b L b Braid relations: [between any lift of A and the inverse of any lift of B] L a L 1 b R a L 1 b Torsion relation: L a = L 1 b L a L 1 R a = L 1 b R a L 1 (L a L 1 b (R a R 1 b b, L a R 1 b b, R a R 1 b L a = R 1 b L a R 1 b, R a) 4 = (L 1 b R ar 1 b )4 = 1, L a) 4 = (R 1 b L al 1 b )4 = 1. R a = R 1 b R a R 1 b. (This was the starting point of joint work with Reutenauer)

101 Lifting the braid relations to Aut(F 2 ) Commutation relations: [the lifts of A (or of B) commute] L a R a = R a L a and L b R b = R b L b Braid relations: [between any lift of A and the inverse of any lift of B] L a L 1 b R a L 1 b Torsion relation: L a = L 1 b L a L 1 R a = L 1 b R a L 1 (L a L 1 b (R a R 1 b b, L a R 1 b b, R a R 1 b L a = R 1 b L a R 1 b, R a) 4 = (L 1 b R ar 1 b )4 = 1, L a) 4 = (R 1 b L al 1 b )4 = 1. R a = R 1 b R a R 1 b. (This was the starting point of joint work with Reutenauer)

102 Lifting the braid relations to Aut(F 2 ) Commutation relations: [the lifts of A (or of B) commute] L a R a = R a L a and L b R b = R b L b Braid relations: [between any lift of A and the inverse of any lift of B] L a L 1 b R a L 1 b Torsion relation: L a = L 1 b L a L 1 R a = L 1 b R a L 1 (L a L 1 b (R a R 1 b b, L a R 1 b b, R a R 1 b L a = R 1 b L a R 1 b, R a) 4 = (L 1 b R ar 1 b )4 = 1, L a) 4 = (R 1 b L al 1 b )4 = 1. R a = R 1 b R a R 1 b. (This was the starting point of joint work with Reutenauer)

103 IV. BRAIDS

104 The braid group B n on n strands Presentation of B n : Generators: σ 1,..., σ n 1 Defining relations: σ i σ j = σ j σ i if i j 2 σ i σ i+1 σ i = σ i+1 σ i σ i+1

105 Generators of the braid group on n strands 1 2 i i + 1 n σ i 1 2 i i + 1 n σ 1 i

106 Relation between B 4 and Aut(F 2 ) Group homomorphism f : B 4 Aut(F 2 ) defined by f (σ 1 ) = L a, f (σ 2 ) = L 1 b, f (σ 3 ) = R a. Theorem. The following sequence is exact: 1 Z 4 B 4 f Aut(F 2 ) Z/2 0 Z 4 = center of B 4, infinite cyclic generated by (σ 1 σ 2 σ 3 ) 4

107 Relation between B 4 and Aut(F 2 ) Group homomorphism f : B 4 Aut(F 2 ) defined by f (σ 1 ) = L a, f (σ 2 ) = L 1 b, f (σ 3 ) = R a. Theorem. The following sequence is exact: 1 Z 4 B 4 f Aut(F 2 ) Z/2 0 Z 4 = center of B 4, infinite cyclic generated by (σ 1 σ 2 σ 3 ) 4

108 The complete picture Theorem. There is a map of exact sequences 1 Z 4 B 4 f Aut(F 2 ) det Z/2 1 = π π = 1 2Z 3 B 3 GL 2 (Z) det Z/2 1 B 3 = A, B AB 1 A = B 1 AB 1 2Z 3 subgroup of center of B 3, generated by (AB 1 A) 4 π : B 4 B 3 defined by π (σ 1 ) = π (σ 3 ) = A and π (σ 2 ) = B 1 Z 4 = center of B 4, generated by (σ 1 σ 2 σ 3 ) 4

109 The special Sturmian monoid St 0 Recall: Monoid St generated by E, L a, R a or by E, L b, R b Definition St 0 = {ϕ St det(ϕ) = 1} Aut(F 2 ) The substitutions L a, R a, L b, R b belong to St 0 The monoid St 0 is generated by L a, R a, L b, R b

110 The special Sturmian monoid St 0 Recall: Monoid St generated by E, L a, R a or by E, L b, R b Definition St 0 = {ϕ St det(ϕ) = 1} Aut(F 2 ) The substitutions L a, R a, L b, R b belong to St 0 The monoid St 0 is generated by L a, R a, L b, R b

111 The special Sturmian monoid St 0 Recall: Monoid St generated by E, L a, R a or by E, L b, R b Definition St 0 = {ϕ St det(ϕ) = 1} Aut(F 2 ) The substitutions L a, R a, L b, R b belong to St 0 The monoid St 0 is generated by L a, R a, L b, R b

112 The special Sturmian monoid St 0 Recall: Monoid St generated by E, L a, R a or by E, L b, R b Definition St 0 = {ϕ St det(ϕ) = 1} Aut(F 2 ) The substitutions L a, R a, L b, R b belong to St 0 The monoid St 0 is generated by L a, R a, L b, R b

113 Presentation of the monoid St 0 Theorem. The monoid St 0 has the following presentation: Generators: L a, R a, L b, R b Defining relations: L a R a = R a L a, L b R b = R b L b, and L a L k b R a = R a Rb k L a, L b L k a R b = R b Ra k L b for all k 1. (This presentation is infinite)

114 St 0 is a submonoid of B 4 The monoid St 0 Aut(F 2 ) can be lifted to a monoid in B 4 Theorem. There is a monoid morphism i : St 0 B 4 such that i f St 0 B 4 Aut(F 2 ) is the inclusion. The monoid embedding i : St 0 B 4 is defined by i(l a ) = σ 1, i(l b ) = σ 1 2, i(r a ) = σ 3, i(r b ) = σ 1 4. What is the braid σ 1 4?

115 St 0 is a submonoid of B 4 The monoid St 0 Aut(F 2 ) can be lifted to a monoid in B 4 Theorem. There is a monoid morphism i : St 0 B 4 such that i f St 0 B 4 Aut(F 2 ) is the inclusion. The monoid embedding i : St 0 B 4 is defined by i(l a ) = σ 1, i(l b ) = σ 1 2, i(r a ) = σ 3, i(r b ) = σ 1 4. What is the braid σ 1 4?

116 St 0 is a submonoid of B 4 The monoid St 0 Aut(F 2 ) can be lifted to a monoid in B 4 Theorem. There is a monoid morphism i : St 0 B 4 such that i f St 0 B 4 Aut(F 2 ) is the inclusion. The monoid embedding i : St 0 B 4 is defined by i(l a ) = σ 1, i(l b ) = σ 1 2, i(r a ) = σ 3, i(r b ) = σ 1 4. What is the braid σ 1 4?

117 The braids σ 1, σ 1 2, σ 3 σ 1 σ 1 2 σ 3

118 The braid σ 1 4 It braids the 1st and the 4th strands with a negative crossing behind the 2nd and 3rd strands σ 1 4 = (σ 1 σ 1 3 ) σ 2 (σ 1 σ 1 3 ) 1

119 Conclusion The monoid generated by σ 1, σ 1 2, σ 3, σ 1 4 in B 4 is isomorphic to the special Sturmian monoid St 0 σ 1, σ 1 2, σ 3, σ = St0 Note: The subgroup generated by σ 1, σ 1 2, σ 3, σ 1 4 is B 4

120 Conclusion The monoid generated by σ 1, σ 1 2, σ 3, σ 1 4 in B 4 is isomorphic to the special Sturmian monoid St 0 σ 1, σ 1 2, σ 3, σ = St0 Note: The subgroup generated by σ 1, σ 1 2, σ 3, σ 1 4 is B 4

121 References - Old and... J. Bernoulli, Sur une nouvelle espèce de calcul. In: Recueil pour les astronomes, t. 1. Berlin (1772), E. B. Christoffel, Observatio arithmetica. Ann. Mat. Pura Appl. 6 (1875), A. Markoff, Sur une question de Jean Bernouilli. Math. Ann. 19 (1882), H. J. S. Smith, Note on continued fractions. Messenger of Mathematics (1876)

122 References - New J.-P. Allouche, J. Shallit, Automatic sequences. Theory, applications, generalizations. Cambridge: Cambridge University Press 2003 C. Kassel, C. Reutenauer, Sturmian morphisms, the braid group B 4, Christoffel words, and bases of F 2. Ann. Mat. Pura Appl. 186 (2007), M. Lothaire, Algebraic combinatorics on words. Cambridge: Cambridge University Press, 2002 R. P. Osborne, H. Zieschang, Primitives in the free group on two generators. Invent. Math. 63 (1981), 17 24

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