{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:PZPMFT4MNXATSGT5EKJZ3STW75","short_pith_number":"pith:PZPMFT4M","schema_version":"1.0","canonical_sha256":"7e5ec2cf8c6dc1391a7d22939dca76ff44ab6790d6980e07e2b086af594c6dcc","source":{"kind":"arxiv","id":"1803.00115","version":1},"attestation_state":"computed","paper":{"title":"Holomorphic quadratic differentials on graphs and the chromatic polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Richard Kenyon, Wai Yeung Lam","submitted_at":"2018-02-28T22:28:03Z","abstract_excerpt":"We study \"holomorphic quadratic differentials\" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $\\chi$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $\\chi(-k)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1803.00115","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-28T22:28:03Z","cross_cats_sorted":[],"title_canon_sha256":"386474c6bfb845fc07c8ec5e4e033ddbc8f1c170b853795dfd4e44e1ea7591e2","abstract_canon_sha256":"9c0ded4b9ef6843ff0e16a3f43a956fbb4e2a17609b50ce84f01f1513d318403"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:13.307935Z","signature_b64":"s96UeG3+Zotit9xk8J+kyi4dDciGXQc1lcbXugrw/6oOPf+SFB5nn5y+noXqdo2psk8si2/wjG52FrkcCay8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7e5ec2cf8c6dc1391a7d22939dca76ff44ab6790d6980e07e2b086af594c6dcc","last_reissued_at":"2026-05-18T00:22:13.307187Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:13.307187Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Holomorphic quadratic differentials on graphs and the chromatic polynomial","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Richard Kenyon, Wai Yeung Lam","submitted_at":"2018-02-28T22:28:03Z","abstract_excerpt":"We study \"holomorphic quadratic differentials\" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $\\chi$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $\\chi(-k)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.00115","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1803.00115","created_at":"2026-05-18T00:22:13.307293+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.00115v1","created_at":"2026-05-18T00:22:13.307293+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.00115","created_at":"2026-05-18T00:22:13.307293+00:00"},{"alias_kind":"pith_short_12","alias_value":"PZPMFT4MNXAT","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_16","alias_value":"PZPMFT4MNXATSGT5","created_at":"2026-05-18T12:32:46.962924+00:00"},{"alias_kind":"pith_short_8","alias_value":"PZPMFT4M","created_at":"2026-05-18T12:32:46.962924+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75","json":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75.json","graph_json":"https://pith.science/api/pith-number/PZPMFT4MNXATSGT5EKJZ3STW75/graph.json","events_json":"https://pith.science/api/pith-number/PZPMFT4MNXATSGT5EKJZ3STW75/events.json","paper":"https://pith.science/paper/PZPMFT4M"},"agent_actions":{"view_html":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75","download_json":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75.json","view_paper":"https://pith.science/paper/PZPMFT4M","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.00115&json=true","fetch_graph":"https://pith.science/api/pith-number/PZPMFT4MNXATSGT5EKJZ3STW75/graph.json","fetch_events":"https://pith.science/api/pith-number/PZPMFT4MNXATSGT5EKJZ3STW75/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75/action/storage_attestation","attest_author":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75/action/author_attestation","sign_citation":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75/action/citation_signature","submit_replication":"https://pith.science/pith/PZPMFT4MNXATSGT5EKJZ3STW75/action/replication_record"}},"created_at":"2026-05-18T00:22:13.307293+00:00","updated_at":"2026-05-18T00:22:13.307293+00:00"}