{"paper":{"title":"Knots, Perturbative Series and Quantum Modularity","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["hep-th"],"primary_cat":"math.GT","authors_text":"Don Zagier, Stavros Garoufalidis","submitted_at":"2021-11-12T10:24:26Z","abstract_excerpt":"We introduce an invariant of a hyperbolic knot which is a map $\\alpha\\mapsto \\boldsymbol{\\Phi}_\\alpha(h)$ from $\\mathbb{Q}/\\mathbb{Z}$ to matrices with entries in $\\overline{\\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\\rm SL}_2(\\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $(\\sigma_0,\\sigma_1)$ entry, where $\\sigma_0$ is the trivial and $\\sigma_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\\rm e}^{2\\pi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.06645","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/2111.06645/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"}