{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:5SQDU6TKVNKACISCI5G4H62HMJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"133bc013af87b09bf6d1e1841c8231419a1a8c7fad07cacacfe3419645d1ceab","cross_cats_sorted":["math.GT","math.QA","math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-10-26T15:21:02Z","title_canon_sha256":"780deee42a4dd6cc88e8b780adda9ec2ab189df716440773cf5466758c285614"},"schema_version":"1.0","source":{"id":"2110.13768","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.13768","created_at":"2026-07-05T04:16:15Z"},{"alias_kind":"arxiv_version","alias_value":"2110.13768v1","created_at":"2026-07-05T04:16:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.13768","created_at":"2026-07-05T04:16:15Z"},{"alias_kind":"pith_short_12","alias_value":"5SQDU6TKVNKA","created_at":"2026-07-05T04:16:15Z"},{"alias_kind":"pith_short_16","alias_value":"5SQDU6TKVNKACISC","created_at":"2026-07-05T04:16:15Z"},{"alias_kind":"pith_short_8","alias_value":"5SQDU6TK","created_at":"2026-07-05T04:16:15Z"}],"graph_snapshots":[{"event_id":"sha256:ff826819065315e682fe2978508d105f8c5ff0c1f6e22a13df36f9da58dbf95c","target":"graph","created_at":"2026-07-05T04:16:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2110.13768/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We generalize the $F_K$ invariant, i.e. $\\widehat{Z}$ for the complement of a knot $K$ in the 3-sphere, the knots-quivers correspondence, and $A$-polynomials of knots, and find several interconnections between them. We associate an $F_K$ invariant to any branch of the $A$-polynomial of $K$ and we work out explicit expressions for several simple knots. We show that these $F_K$ invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using $R$-matrices. We ","authors_text":"Angus Gruen, Marko Sto\\v{s}i\\'c, Piotr Kucharski, Piotr Su{\\l}kowski, Sergei Gukov, Sunghyuk Park, Tobias Ekholm","cross_cats":["math.GT","math.QA","math.SG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-10-26T15:21:02Z","title":"Branches, quivers, and ideals for knot complements"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.13768","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:74e6ad2fa9d18b36ee95c6bd99df285110f6013b2e58efaa0e6cfed3a71e79f1","target":"record","created_at":"2026-07-05T04:16:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"133bc013af87b09bf6d1e1841c8231419a1a8c7fad07cacacfe3419645d1ceab","cross_cats_sorted":["math.GT","math.QA","math.SG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-10-26T15:21:02Z","title_canon_sha256":"780deee42a4dd6cc88e8b780adda9ec2ab189df716440773cf5466758c285614"},"schema_version":"1.0","source":{"id":"2110.13768","kind":"arxiv","version":1}},"canonical_sha256":"eca03a7a6aab54012242474dc3fb47625a833030ca21bb47beff204d6f060153","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eca03a7a6aab54012242474dc3fb47625a833030ca21bb47beff204d6f060153","first_computed_at":"2026-07-05T04:16:15.925093Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:16:15.925093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7dsvVDTKjuXfP9364/hkxoGUuK+VUwNmsMRT8X6Z1rD2CSzX+z4m6afeMR8cUa+sRm2ejR4NzGjePjTPMiC+Ag==","signature_status":"signed_v1","signed_at":"2026-07-05T04:16:15.925681Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.13768","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:74e6ad2fa9d18b36ee95c6bd99df285110f6013b2e58efaa0e6cfed3a71e79f1","sha256:ff826819065315e682fe2978508d105f8c5ff0c1f6e22a13df36f9da58dbf95c"],"state_sha256":"a3538cb7145d6951eef92757b68ddb15a061bc6955efe42e9c94be7a1a4c6230"}