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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2106.15886","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by-sa/4.0/","primary_cat":"math.CO","submitted_at":"2021-06-30T08:22:30Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"d881a21b2c4b8c96660c2fc3aa4ae9932c993ed5e71ef48732040fc0e56d2102","abstract_canon_sha256":"c576b862507ef43805b89a4b469160a0a647b1289af94856ec7961eaff1f2279"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:14:37.870420Z","signature_b64":"Iw7eIWr0rBKBIola9oVz6i7gy1GtU6RrBR1bcBb9vckE+tFOUNl4nsFuO4Y1Bro9Z/9Wdx7lEIhmsuRRBAckBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"829e5922a280e6ffb397f751bfbe8fd1f63feb10ea5fe9e07a1d572d0fa5c5ed","last_reissued_at":"2026-07-05T04:14:37.869983Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:14:37.869983Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $q=1$. 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