Geodesic
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, Tome 6 (2023) no. 6, pp. 1677-1685

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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.

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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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {A $q$-analog of the {Markoff} injectivity conjecture holds},
     journal = {Algebraic Combinatorics},
     pages = {1677--1685},
     publisher = {The Combinatorics Consortium},
     volume = {6},
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     year = {2023},
     doi = {10.5802/alco.322},
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Labbé, Sébastien; Lapointe, Mélodie; Steiner, Wolfgang. A $q$-analog of the Markoff injectivity conjecture holds. Algebraic Combinatorics, Tome 6 (2023) no. 6, pp. 1677-1685. doi: 10.5802/alco.322

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