{"paper":{"title":"The Tangle Hypothesis: Dimension 1","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CT","math.GT","math.QA"],"primary_cat":"math.AT","authors_text":"David Ayala, John Francis","submitted_at":"2024-10-31T14:18:40Z","abstract_excerpt":"We introduce an $(\\infty,1)$-category ${\\sf Bord}_1^{\\sf fr}(\\mathbb{R}^n)$, the morphisms in which are framed tangles in $\\mathbb{R}^n\\times \\mathbb{D}^1$. We prove that ${\\sf Bord}_1^{\\sf fr}(\\mathbb{R}^n)$ has the universal mapping out property of the 1-dimensional Tangle Hypothesis of Baez--Dolan and Hopkins--Lurie: it is the rigid $\\mathcal{E}_n$-monoidal $(\\infty,1)$-category freely generated by a single object. Applying this theorem to a dualizable object of a braided monoidal $(\\infty,1)$-category gives link invariants, generalizing the Reshetikhin--Turaev invariants."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.23965","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/2410.23965/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"}