ALGEBRAIC COMBINATORICS

no. 6
A q-analog of the Markoff injectivity conjecture holds
Labbé, Sébastien1; Lapointe, Mélodie2; Steiner, Wolfgang3
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
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.

Received:
Revised:
Accepted:
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 
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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/

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