A q-analog of the Markoff injectivity conjecture holds
Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1677-1685.

The elements of Markoff triples are given by coefficients in certain matrix products defined by Christoffel words, and the Markoff injectivity conjecture, a longstanding open problem (also known as the uniqueness conjecture), is then equivalent to injectivity on Christoffel words. A q-analog of these matrix products has been proposed recently, and we prove that injectivity on Christoffel words holds for this q-analog. The proof is based on the evaluation at q=exp(iπ/3). Other roots of unity provide some information on the original problem, which corresponds to the case q=1. We also extend the problem to arbitrary words and provide a large family of pairs of words where injectivity does not hold.

Published online:
DOI: 10.5802/alco.322
Classification: 11J06, 68R15, 05A30
Keywords: Markoff number, Christoffel word, $q$-analog
Labbé, Sébastien 1; Lapointe, Mélodie 2; Steiner, Wolfgang 3

1 Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France
2 Université de Moncton Département de mathématiques et de statistique 18 avenue Antonine-Maillet Moncton NB E1A 3E9, Canada
3 Université Paris Cité, CNRS, IRIF, F–75006 Paris, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Labb\'e, S\'ebastien and Lapointe, M\'elodie and Steiner, Wolfgang},
     title = {A $q$-analog of the {Markoff} injectivity conjecture holds},
     journal = {Algebraic Combinatorics},
     pages = {1677--1685},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {6},
     year = {2023},
     doi = {10.5802/alco.322},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.322/}
AU  - Labbé, Sébastien
AU  - Lapointe, Mélodie
AU  - Steiner, Wolfgang
TI  - A $q$-analog of the Markoff injectivity conjecture holds
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1677
EP  - 1685
VL  - 6
IS  - 6
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.322/
DO  - 10.5802/alco.322
LA  - en
ID  - ALCO_2023__6_6_1677_0
ER  - 
%0 Journal Article
%A Labbé, Sébastien
%A Lapointe, Mélodie
%A Steiner, Wolfgang
%T A $q$-analog of the Markoff injectivity conjecture holds
%J Algebraic Combinatorics
%D 2023
%P 1677-1685
%V 6
%N 6
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.322/
%R 10.5802/alco.322
%G en
%F ALCO_2023__6_6_1677_0
Labbé, Sébastien; Lapointe, Mélodie; Steiner, Wolfgang. A $q$-analog of the Markoff injectivity conjecture holds. Algebraic Combinatorics, Volume 6 (2023) no. 6, pp. 1677-1685. doi : 10.5802/alco.322. https://alco.centre-mersenne.org/articles/10.5802/alco.322/

[1] Aigner, Martin Markov’s theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings, Springer, Cham, 2013, x+257 pages | DOI | MR

[2] Bazier-Matte, Véronique; Schiffler, Ralf Knot theory and cluster algebras, Adv. Math., Volume 408 (2022), Paper no. 108609, 45 pages | DOI | MR | Zbl

[3] Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. Combinatorics on words. Christoffel words and repetitions in words, CRM Monograph Series, 27, American Mathematical Society, Providence, RI, 2009, xii+147 pages | MR

[4] Cotti, Giordano; Varchenko, Alexander The *-Markov equation for Laurent polynomials, Mosc. Math. J., Volume 22 (2022) no. 1, pp. 1-68 | DOI | MR | Zbl

[5] Frobenius, G. Über die Markoffschen Zahlen, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Volume 26 (1913), pp. 458-487 | Zbl

[6] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren Concrete mathematics. A foundation for computer science, Addison-Wesley Publishing Company, Reading, MA, 1994, xiv+657 pages | MR

[7] Kantarcı Oğuz, Ezgi; Ravichandran, Mohan Rank polynomials of fence posets are unimodal, Discrete Math., Volume 346 (2023) no. 2, Paper no. 113218, 20 pages | DOI | MR | Zbl

[8] Kogiso, Takeyoshi q-Deformations and t-deformations of Markov triples (2020) (Accessed 2023-06-07) | arXiv

[9] Labbé, Sébastien; Lapointe, Mélodie The q-analog of the Markoff injectivity conjecture over the language of a balanced sequence, Comb. Theory, Volume 2 (2022) no. 1, Paper no. 9, 25 pages | DOI | MR | Zbl

[10] Lapointe, Mélodie; Reutenauer, Christophe On the Frobenius conjecture, Integers, Volume 21 (2021), Paper no. A67, 9 pages | MR | Zbl

[11] Leclere, Ludivine; Morier-Genoud, Sophie q-deformations in the modular group and of the real quadratic irrational numbers, Adv. in Appl. Math., Volume 130 (2021), Paper no. 102223, 28 pages | DOI | MR | Zbl

[12] Markoff, A. Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 15 (1879) no. 3, pp. 381-406 | DOI | Zbl

[13] Markoff, A. Sur les formes quadratiques binaires indéfinies (second mémoire), Math. Ann., Volume 17 (1880) no. 3, pp. 379-399 | DOI | MR | Zbl

[14] McConville, Thomas; Sagan, Bruce E.; Smyth, Clifford On a rank-unimodality conjecture of Morier-Genoud and Ovsienko, Discrete Math., Volume 344 (2021) no. 8, Paper no. 112483, 13 pages | DOI | MR | Zbl

[15] Morier-Genoud, Sophie; Ovsienko, Valentin q-deformed rationals and q-continued fractions, Forum Math. Sigma, Volume 8 (2020), Paper no. e13, 55 pages | DOI | MR | Zbl

[16] OEIS Foundation Inc. Entry A002559 in The On-Line Encyclopedia of Integer Sequences, 2022 http://oeis.org/A002559

[17] Ovsienko, Valentin Towards quantized complex numbers: q-deformed Gaussian integers and the Picard group, Open Communications in Nonlinear Mathematical Physics, Volume Volume 1 (2021) | DOI

[18] Reutenauer, Christophe Christoffel words and Markoff triples, Integers, Volume 9 (2009), pp. A26, 327-332 | DOI | MR | Zbl

[19] Reutenauer, Christophe From Christoffel words to Markoff numbers, Oxford University Press, Oxford, 2019, xi+156 pages | MR

Cited by Sources: