{"paper":{"title":"Wug-snake graphs and Markov numbers of matrix semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Generalized Markov numbers from semigroups of reduced integer matrices are given by perfect matchings of wug-snake graphs.","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Oleg Karpenkov, Yefei Ma","submitted_at":"2026-04-18T17:06:37Z","abstract_excerpt":"Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence to matrix semigroups by introducing weighted universal generalised snake graphs, or wug-snake graphs for short. These are weighted bipartite graphs whose perfect matching sequences encode linear recurrences. We associate to each wug-snake graph a continuant matrix and prove that its determinant equals the weighted perfect matching number. We use this construction to define polymino wug-tiles for matrices and show that their determinants compute Markov-Davenport forms. Con"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Generalized Markov numbers arising from semigroups of reduced integer matrices can be found by counting perfect matchings of wug-snake graphs.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the specific semigroup construction on reduced integer matrices produces objects that qualify as generalized Markov numbers and that the perfect-matching count on the associated wug-snake graphs exactly equals those numbers.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Markov numbers from matrix semigroups equal the number of perfect matchings in wug-snake graphs.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Markov numbers from semigroups of reduced integer matrices are given by perfect matchings of wug-snake graphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e0816e51b6a487ee9f4135bc6f5ae37a6ee9d455eb83d9da6344afd7daad6395"},"source":{"id":"2604.17069","kind":"arxiv","version":2},"verdict":{"id":"45e1ca20-e8e2-40ff-9f6e-ac560729b668","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T06:22:19.149793Z","strongest_claim":"Generalized Markov numbers arising from semigroups of reduced integer matrices can be found by counting perfect matchings of wug-snake graphs.","one_line_summary":"Generalized Markov numbers from matrix semigroups equal the number of perfect matchings in wug-snake graphs.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the specific semigroup construction on reduced integer matrices produces objects that qualify as generalized Markov numbers and that the perfect-matching count on the associated wug-snake graphs exactly equals those numbers.","pith_extraction_headline":"Generalized Markov numbers from semigroups of reduced integer matrices are given by perfect matchings of wug-snake graphs."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.17069/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"}