Markov numbers

Some (mostly on-line) resources

Number Theoretic properties

• Don Zagier, On the Number of Markoff Numbers Below a Given Bound.
• Pietro Corvaja, Umberto Zannier, On the Greatest Prime Factor of Markov Pairs.

Uniqueness conjecture:

• Markov's Theorem and 100 Years of the Uniqueness Conjecture. A Mathematical Journey from Irrational Numbers to Perfect Matchings by Martin Aigner.
• D.Rosen, G.S.Patterson, Some Numerical Evidence Concerning the Uniqueness of the Markov Numbers.
• I. Borosh, More numberical evidence on the uniqueness of Markov numbers.
• K.R. Matthews, J.P. Robertson, An equivalent form of the unicity conjecture for Markoff numbers.
• Ying Zhang, Congruence and uniqueness of certain Markoff numbers .
• Ying Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers .
• Mong Lung Lang, Ser Peow Tan, A simple proof of the Markoff conjecture for prime powers.

Geometry of Markov numbers:

• Caroline Series, The Geometry of Markoff Numbers.
• Yi Huang, Notes for "The Geometry of Markoff Numbers” by Caroline Series.
• Yi Huang, Paul Norbury, Simple geodesics and Markoff quads.
• Caroline Series, Symbolic Dynamics for Geodesic Flows.
• A.F.Beardon, J.Lehner, M.Sheingorn, Closed geodesics on a riemann surface with application to the Markov spectrum.
• Greg McShane, Hugo Parlier, Simple closed geodesics of equal length on a torus .
• Andrew Haas, The geometry of Markoff forms.
• Hervey Cohn, Approach to Markoff's minimal forms through modular functions.

Continued Fractions

• Oleg Karpenkov, Geometry of Continued Fractions.
• Caroline Series, Continued Fractions and Hyperbolic Geometry.
• Wieb Bosma, Cor Kraaikamp (and 9 students), Continued Fractions, notes of a course.
• Enrico Bombieri, Continued fractions and the Markoff tree.
• Ian Short, The hyperbolic geometry of continued fractions.

Farey and Euclid trees:

• Farey sequences, the Farey map and the Farey Tree webpage by Matthew R. Watkins.
• Donald Joseph Mcginn, Generalized Markoff Equations, Euclid Trees, and Chebyshev Polynomials.
• Sophie Morier-Genoud, Valentin Ovsienko, Serge Tabachnikov, SL_2(Z)-tilings of the torus, Coxeter-Conway friezes and Farey triangualtions .
• Jonathan Ainsworth, Michael Dawson, John Pianta, James Warwick, The Farey sequence, Report on 4H project, Edinbourgh, 2012.

Combinatorics of Markov numbers

• Christophe Reutenauer, Christofel words and Markoff triples.
• James Propp, The combinatorics of frieze patterns and Markoff numbers.
• Andy Itsara, Gregg Musiker, James Propp, and Rui Viana, Combinatorial Interpretations for the Markov Numbers.

Markov numbers and cluster algebras

• Xueyuan Peng, Jie Zhang, Cluster algebras and Markoff numbers.
• Philipp Lampe, Diophantine equations via cluster transformations .
• Ilke Canakci, Ralf Schiffler, Cluster algebras and continued fractions.

Introduction to cluster algebras:

• Philipp Lampe, Cluster algebras.
• Lauren Williams, Cluster algebras: an introduction.
• Sergey Fomin, Lauren Williams, Andrei Zelevinsky, Introduction to Cluster Algebras, Chapters 1–3.

Markov equation, generalised Markov equation:

• Sara Maloni, Frederic Palesi, Ser Peow Tan, On the character variety of the hour-holed sphere.
• Andre Beineke, Thomas Brustle, Lutz Hille, Cluster-Cyclic Quivers with three Vertices and the Markov Equation, with an appendix by Otto Kerner .

Some webpages:

• Markov numbers by Tom Ace.
• Markov numbers, Wikipedia

Slides of some talks:

• Markov numbers by Kha H. Man.