{"paper":{"title":"Modular knots, automorphic forms, and the Rademacher symbols for triangle groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GT","authors_text":"Jun Ueki, Toshiki Matsusaka","submitted_at":"2021-09-02T17:39:57Z","abstract_excerpt":"\\'{E}.\\~Ghys proved that the linking numbers of modular knots and the \"missing\" trefoil $K_{2,3}$ in $S^3$ coincide with the values of a highly ubiquitous function called the Rademacher symbol for ${\\rm SL}_2\\mathbb{Z}$. In this paper, we replace ${\\rm SL}_2\\mathbb{Z}=\\Gamma_{2,3}$ by the triangle group $\\Gamma_{p,q}$ for any coprime pair $(p,q)$ of integers with $2\\leq p<q$. We invoke the theory of harmonic Maass forms for $\\Gamma_{p,q}$ to introduce the notion of the Rademacher symbol $\\psi_{p,q}$, and provide several characterizations. Among other things, we generalize Ghys's theorem for mo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.01114","kind":"arxiv","version":2},"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/2109.01114/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"}