{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GHBBLF2XXKY5ADGSPYMZN2IYHA","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":"f2bb36a812043e6114cd97e3dbf55fd0b0c0bf630a60076f1eaf969f68ce64d3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-13T09:58:30Z","title_canon_sha256":"9baeadcda88a5c7a0ff46d55be653024e74655c104f2868f2f1c4256d88e3ab7"},"schema_version":"1.0","source":{"id":"1105.2675","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1105.2675","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"arxiv_version","alias_value":"1105.2675v2","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.2675","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"pith_short_12","alias_value":"GHBBLF2XXKY5","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"GHBBLF2XXKY5ADGS","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"GHBBLF2X","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:74ffdd221436e7a35353b87fde0ebeb3b720966c2a5b34a236ff1f2b71b00113","target":"graph","created_at":"2026-05-18T03:21:32Z","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"},"paper":{"abstract_excerpt":"We introduce a modular (integral) complementary polynomial $\\kappa(G;x,y)$ ($\\kappa_{\\mathbbm z}(G;x,y)$) of two variables of a graph $G$ by counting the number of modular (integral) complementary tension-flows (CTF) of $G$ with an orientation $\\epsilon$. We study these polynomials by further introducing a cut-Eulerian equivalence relation on orientations and geometric structures such as the complementary open lattice polyhedron $\\Delta_\\textsc{ctf}(G,\\epsilon)$, the complementary open 0-1 polytope $\\Delta^+_\\textsc{ctf}(G,\\epsilon)$, and the complementary open lattice polytopes $\\Delta^\\rho_\\","authors_text":"Beifang Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-13T09:58:30Z","title":"Dual complementary polynomials of graphs and combinatorial interpretation on the values of the Tutte polynomial at positive integers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2675","kind":"arxiv","version":2},"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":"sha256:118053e00da2d277cb70fa85c43d755a471f050193f712fcbbd6e186f5c0148f","target":"record","created_at":"2026-05-18T03:21:32Z","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":"f2bb36a812043e6114cd97e3dbf55fd0b0c0bf630a60076f1eaf969f68ce64d3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-13T09:58:30Z","title_canon_sha256":"9baeadcda88a5c7a0ff46d55be653024e74655c104f2868f2f1c4256d88e3ab7"},"schema_version":"1.0","source":{"id":"1105.2675","kind":"arxiv","version":2}},"canonical_sha256":"31c2159757bab1d00cd27e1996e9183810ed556d131180afc1bf8c5b66513d4f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"31c2159757bab1d00cd27e1996e9183810ed556d131180afc1bf8c5b66513d4f","first_computed_at":"2026-05-18T03:21:32.050105Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:32.050105Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"O8mbXBJ6vTKvLtCQl3ktfJLIa1ugAs+YlB7nNSu9qFbZwyddD27XsgNDdkaTo/u3ZiST2k4Y6J0zJtLrlQGaBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:32.050558Z","signed_message":"canonical_sha256_bytes"},"source_id":"1105.2675","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:118053e00da2d277cb70fa85c43d755a471f050193f712fcbbd6e186f5c0148f","sha256:74ffdd221436e7a35353b87fde0ebeb3b720966c2a5b34a236ff1f2b71b00113"],"state_sha256":"793b0cda06d8b4439b53dd3ad70650cea421ee0d821cbbcf7a4705fb5376a6ca"}