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Extended Markov Numbers and Integer Geometry Extended Markov numbers and integer geometry Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Matty van Son March 2020 Abstract Extended Markov numbers and integer geometry Matty van Son We call two lines intersecting at the point (0; 0) in R2 an arrangement. Arrange- ments are solutions to equations ax2 + bxy + cy2 = 0; for real a, b, and c satisfying b2 − 4ac > 0. Solutions to the Markov Diophantine equation x2 + y2 + z2 = 3xyz are called Markov numbers, and are a key component in classifying 2-D arrange- ments up to SL(2; Z) equivalence, as shown by A. Markov in the late 19th century. In this thesis we prove the following results: (i) There is a relation called the Perron identity between infinite sequences of positive integers and the minima of binary quadratic forms. We find the geometric meaning behind this result, and extend it to all values of forms in Theorem 2.3.14; (ii) Markov numbers may be described through combinations of the sequences (1; 1) and (2; 2). These combinations are known either as Cohn words or Christoffel words. We propose an extension to the theory of Markov num- bers. The initial sequences used in the classical construction are substituted by new ones with certain properties. In Theorems 4.1.14, 4.1.27, and 5.1.5 we show that these sequences create systems of numbers similar to Markov numbers; (iii) There is a conjecture first noted by G. Frobenius in 1913 that says triples of Markov numbers are defined by their largest element. We see analogues to this uniqueness conjecture for each extension of Markov numbers. We show that for certain extensions of Markov numbers, the generalised uniqueness conjecture fails in Theorem 5.2.8. 2 Declaration No part of the work in this thesis has been submitted for a degree or other qualification at this or any other institution. Material in Section 2.3 appears in the following paper. [54] O. Karpenkov and M. Van Son. Generalized Perron Identity for broken lines. Journal de th´eoriedes nombres de Bordeaux, 31:131{144, 2019. 3 Contents List of Figures 5 List of important definitions 8 List of maps between triple graphs 9 1 Introduction 10 1.1 History of regular Markov numbers . 11 1.1.1 Markov's theorem . 11 1.1.2 Graph of regular Markov numbers . 14 1.1.3 Recent work . 17 1.2 Motivating example for integer geometry . 18 1.3 Background of integer geometry . 20 1.3.1 Continued fractions . 20 1.3.2 Concepts of integer geometry . 22 1.3.3 Perron identity . 27 1.4 Research layout . 29 2 Background and definitions 32 2.1 Continuants . 33 2.2 A note on SL(2; Z) transformations . 36 2.2.1 SL(2; Z) invariants . 36 2.2.2 SL(2; Z) transformations of forms . 41 2.3 Integer geometry and the Markov spectrum . 51 2.4 Proof of Theorem 2.3.14 . 59 2.5 Markov spectrum and Markov forms . 63 2.5.1 Markov spectrum and forms . 63 2.5.2 Cohn matrices . 65 3 Triple graph structure of Markov numbers 68 3.1 Triple graph structure . 69 3.2 Ternary operations . 71 4 CONTENTS 5 3.2.1 Triple operation for regular Markov sequences . 71 3.2.2 Regular reduced Markov Matrices . 76 3.2.3 Regular reduced Markov forms . 80 3.2.4 Matrix similarity and form equivalence . 87 3.3 Graph relations . 93 3.3.1 Relation between forms and matrices . 93 3.3.2 Inverse maps . 96 3.4 Diagram of Classical Markov theory . 100 3.4.1 Markov spectrum maps . 101 3.4.2 Perron identity for Markov sequences . 104 3.4.3 Partial inverse of Markov number maps . 107 4 General Markov numbers 108 4.1 General Markov theory . 110 4.1.1 Markov graphs . 110 4.1.2 Minimum condition . 113 4.1.3 Evenly-palindromic Markov graphs . 121 4.2 Proof of Theorem 4.1.27: Palindromic Markov sequences . 125 4.2.1 Alternative definition for S(n)................ 126 4.2.2 Symmetry of construction of sequences S(n) ....... 130 4.2.3 Canonical form for Markov sequences . 135 4.2.4 Proof of Proposition 4.2.2 . 139 4.2.5 Proof of Theorem 4.1.27 . 142 4.3 Markov graph relations . 144 4.3.1 General forms . 144 4.3.2 Matrices . 145 4.3.3 General Markov numbers . 146 4.4 Maps . 149 4.4.1 Inverse maps . 149 4.4.2 Markov spectrum maps . 149 5 General Markov numbers 151 5.1 General Markov numbers . 152 5.2 Uniqueness conjecture . 158 5.2.1 First counterexamples to the general uniqueness conjecture 159 5.2.2 Proof of Theorem 5.2.8 . 160 5.3 Further study . 165 5.3.1 Some experimental data for the general uniqueness conjecture165 5.3.2 Open Questions . 165 5 List of Figures 1.1 The first 4 levels in the tree of regular Markov numbers. 16 1.2 Two lines L1 and L2.......................... 18 1.3 The arrangement defined by the lines L1 and L2........... 19 1.4 The point counting for the integer area of the triangle 4ABC... 23 1.5 Left hand side: some triangles with lS = 1. Right hand side: some triangles with lS = 4. 24 1.6 A sail and an LLS sequence for the form 7xy − 3y2......... 25 1.7 The LLS sequences of the sails for the form f(x; y) = 5x2 +14xy − 10y2. The integer lengths are written in black, the integer sines in white. 27 1.8 The classical structure of Markov theory. 29 1.9 The extended structure of Markov theory. 30 2.1 How orientation shows whether points are on the same half plane or not. 37 2.2 The pentagons P (in black) and MP (in red). 39 2.3 A point P in the interior of a triangle 4A1A2A3.......... 40 + − + − 2.4 The rays L1 , L1 , L2 , and L2 ..................... 41 + − 2.5 Vf and Vf ............................... 42 2.6 The sail for α = (1; 1; 2; 2) being mapped to the sail of α3 = (2; 2; 1; 1). ............................... 49 2.7 The sail for α = (1; 1; 2; 2) being mapped to the sail of α2 = (1; 2; 2; 1). ............................... 49 2.8 Left hand side: A broken line for the form 3y2 − 2xy − 5x2. Right hand side: The sail of the acute angle between 3y = 5x and y =0................................ 52 2.9 The LSS sequence for the sail of the acute angle between 3y = 5x and y =0................................ 53 2.10 The four sails for the form 5x2 + 9xy − 7y2, given by the LSS sequence with period (1; 1; 2; 2). 55 2.11 The form (y − 2x)(3y + x) and broken line A = A0 :::A5...... 56 6 LIST OF FIGURES 7 2.12 Sail and LLS sequence for the form 5x2 + 9xy − 7y2. Integer sines are written in red, and integer lengths are written in black. 57 2.13 The kernel of f and the f-broken line P AQ............. 59 2.14 The original f-broken line A and the resulting f-broken line BAnC. 61 2.15 Structure of the Markov spectrum below 3. 64 2.16 The structure of the first 4 levels in the tree of Cohn matrices with starting pair (A; B). ......................... 66 3.1 The first 4 levels in an arbitrary graph G(S; σ; v). 70 3.2 The first three levels of G⊕ (1; 1); (2; 2) ............... 73 3.3 The first three levels in a graph G⊕(µ, ν). 74 3.4 The first three levels in a graph X G⊕(µ, ν) ............ 74 3.5 The forms from first 3 levels in the graph of regular reduced Markov forms. 85 3.6 Relations between regular sequences and matrices, and regular se- quences and forms. 96 3.7 Relations between regular Markov sequences, matrices, and forms. 98 3.8 Relations between regular Markov sequences, matrices, and forms. 100 3.9 The classical structure of Markov theory. 100 3.10 The classical structure of Markov theory. 101 4.1 The left figure shows the sails for µ = (1; 1) and µν = (1; 1; 10; 10). The right figure shows the corresponding cones for y > 0. 116 4.2 Stitching the sails of µ and ν together to get the sail of µν..... 118 4.3 A triple graph with the level of the vertices written in black, and ordering within the level in red. 122 4.4 The first entries in Stern's diatomic sequence. 131 4.5 The extended structure of Markov theory. 149 4.6 The general structure of Markov theory. 150 5.1 The first four levels in the general Markov graphs T (4; 4); (11; 11)152 5.2 The first four levels in the general Markov graph T (µ, ν) for µ = (1; 1) and ν = (3; 2; 1; 1; 2; 3)..................... 153 7 Important Definitions Perron identity: Regular Perron identity: Theorem 1.3.19, p. 27. General Perron identity: Theorem 2.3.14, p. 55. Reduced forms: Definition 2.2.18, p. 43. Regular reduced Markov forms: Definition 3.2.25, p. 81. General reduced Markov forms: Definition 4.1.6, p. 111. Reduced matrices: Definition 3.2.13, p. 77. Regular reduced Markov matrices: Definition 3.2.16, p. 78. General reduced Markov matrices: Definition 4.3.2, p. 145. Cohn matrices: Definition 2.5.4, p. 65.
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