{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:JOJWG5E2XTTADHPTN4Z2CUBAWV","short_pith_number":"pith:JOJWG5E2","canonical_record":{"source":{"id":"1507.03163","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-11T22:25:33Z","cross_cats_sorted":["hep-th","math.GT"],"title_canon_sha256":"c367b4c7f5d9bf4b8e5918277207457cba0072b96e032d569bb59c37eae48c40","abstract_canon_sha256":"394b2ee101fa712dd89cba0a3130ba393a76917fd1210d9d537c052ae8622242"},"schema_version":"1.0"},"canonical_sha256":"4b9363749abce6019df36f33a15020b56c09632b0a1b20d0fd61621b84a25eb5","source":{"kind":"arxiv","id":"1507.03163","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03163","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03163v3","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03163","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"pith_short_12","alias_value":"JOJWG5E2XTTA","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JOJWG5E2XTTADHPT","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JOJWG5E2","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:JOJWG5E2XTTADHPTN4Z2CUBAWV","target":"record","payload":{"canonical_record":{"source":{"id":"1507.03163","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-11T22:25:33Z","cross_cats_sorted":["hep-th","math.GT"],"title_canon_sha256":"c367b4c7f5d9bf4b8e5918277207457cba0072b96e032d569bb59c37eae48c40","abstract_canon_sha256":"394b2ee101fa712dd89cba0a3130ba393a76917fd1210d9d537c052ae8622242"},"schema_version":"1.0"},"canonical_sha256":"4b9363749abce6019df36f33a15020b56c09632b0a1b20d0fd61621b84a25eb5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:32.543556Z","signature_b64":"FArkUAPQdcCriuTj8Xe62MgPm+5JGdMOyaWfif/bE7PQYmL1GpWQITYGgtyq727HyqchbSmTKPwhnaDNcV5kCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b9363749abce6019df36f33a15020b56c09632b0a1b20d0fd61621b84a25eb5","last_reissued_at":"2026-05-18T01:08:32.543034Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:32.543034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.03163","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8/VIaBDmchdWkus3gxS58oJdJ1/UNve3Ie24hR7bQq4Nmczc1HUtmCf1beoCde6b/8ulWbA3fwjZ6qygw/NDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:57:43.411148Z"},"content_sha256":"c7de55712032c76721a7fc012f3118bcadcb4c33ca3894067cad424cf416866f","schema_version":"1.0","event_id":"sha256:c7de55712032c76721a7fc012f3118bcadcb4c33ca3894067cad424cf416866f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:JOJWG5E2XTTADHPTN4Z2CUBAWV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maps, immersions and permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.GT"],"primary_cat":"math.CO","authors_text":"Jean-Bernard Zuber, Robert Coquereaux","submitted_at":"2015-07-11T22:25:33Z","abstract_excerpt":"We consider the problem of counting and of listing topologically inequivalent \"planar\" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03163","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ti+y3LAl7HGes7y+ybjmGjt5KaaJKTzdsGQRRyz9HFE9PnO/oAzvvY5Xoq1Dml73nHp709EwS22OLTdskz4QBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T18:57:43.411656Z"},"content_sha256":"f00b48c7136921867455c0c890ff956ffb941e16ffa8b9a49a18f4799c8e25b0","schema_version":"1.0","event_id":"sha256:f00b48c7136921867455c0c890ff956ffb941e16ffa8b9a49a18f4799c8e25b0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/bundle.json","state_url":"https://pith.science/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T18:57:43Z","links":{"resolver":"https://pith.science/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV","bundle":"https://pith.science/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/bundle.json","state":"https://pith.science/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JOJWG5E2XTTADHPTN4Z2CUBAWV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JOJWG5E2XTTADHPTN4Z2CUBAWV","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":"394b2ee101fa712dd89cba0a3130ba393a76917fd1210d9d537c052ae8622242","cross_cats_sorted":["hep-th","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-11T22:25:33Z","title_canon_sha256":"c367b4c7f5d9bf4b8e5918277207457cba0072b96e032d569bb59c37eae48c40"},"schema_version":"1.0","source":{"id":"1507.03163","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03163","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03163v3","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03163","created_at":"2026-05-18T01:08:32Z"},{"alias_kind":"pith_short_12","alias_value":"JOJWG5E2XTTA","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JOJWG5E2XTTADHPT","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JOJWG5E2","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:f00b48c7136921867455c0c890ff956ffb941e16ffa8b9a49a18f4799c8e25b0","target":"graph","created_at":"2026-05-18T01:08:32Z","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 consider the problem of counting and of listing topologically inequivalent \"planar\" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options where the circle and/or the sphere are/is oriented are considered in turn, following Arnold's classification of the different types of symmetries. We also consider the case of bicolourable and bicoloured maps or immersions, where faces are bicoloured. Our method extends to immersio","authors_text":"Jean-Bernard Zuber, Robert Coquereaux","cross_cats":["hep-th","math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-11T22:25:33Z","title":"Maps, immersions and permutations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03163","kind":"arxiv","version":3},"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:c7de55712032c76721a7fc012f3118bcadcb4c33ca3894067cad424cf416866f","target":"record","created_at":"2026-05-18T01:08:32Z","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":"394b2ee101fa712dd89cba0a3130ba393a76917fd1210d9d537c052ae8622242","cross_cats_sorted":["hep-th","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-11T22:25:33Z","title_canon_sha256":"c367b4c7f5d9bf4b8e5918277207457cba0072b96e032d569bb59c37eae48c40"},"schema_version":"1.0","source":{"id":"1507.03163","kind":"arxiv","version":3}},"canonical_sha256":"4b9363749abce6019df36f33a15020b56c09632b0a1b20d0fd61621b84a25eb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b9363749abce6019df36f33a15020b56c09632b0a1b20d0fd61621b84a25eb5","first_computed_at":"2026-05-18T01:08:32.543034Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:32.543034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FArkUAPQdcCriuTj8Xe62MgPm+5JGdMOyaWfif/bE7PQYmL1GpWQITYGgtyq727HyqchbSmTKPwhnaDNcV5kCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:32.543556Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.03163","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c7de55712032c76721a7fc012f3118bcadcb4c33ca3894067cad424cf416866f","sha256:f00b48c7136921867455c0c890ff956ffb941e16ffa8b9a49a18f4799c8e25b0"],"state_sha256":"d8f878ea0a34d9bb6452461729cc46b511a1bd5bec6e07d1f305b367936dd18a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0HWSR2k2y9rvU3Lo/VeF9lMTLcX5parGqiok1co7ruofT6Kte5RLwVNbkd252AgynnDBfIRhuZ9qnSg3qLgbAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T18:57:43.414253Z","bundle_sha256":"260a84244b5d7bda46ba031bdb6c6a35f97daf137ff8dcc765bc9835586e58e5"}}