{"paper":{"title":"The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"M\\'elodie Lapointe, S\\'ebastien Labb\\'e","submitted_at":"2021-06-30T08:22:30Z","abstract_excerpt":"The Markoff injectivity conjecture states that $w\\mapsto\\mu(w)_{12}$ is injective on the set of Christoffel words where $\\mu:\\{\\mathtt{0},\\mathtt{1}\\}^*\\to\\mathrm{SL}_2(\\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\\mu_q$ of $\\mu$ such that $\\mu_{q}(w)_{12}|_{q=1}=\\mu(w)_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order $<_{radix}$ on $\\{\\mathtt{0},\\mathtt{1}\\}^*$ such that for every bala"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.15886","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.15886/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}