{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:RY3H7SC724JNSTNO2FRUJFSV3V","short_pith_number":"pith:RY3H7SC7","canonical_record":{"source":{"id":"1112.1809","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-08T11:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"60830ad71f05d973b10cf2392c5a5cdf9807c4d47d9b707d1b25484305175347","abstract_canon_sha256":"15e60e743dc5af72e49b7412850d8fd18757f8baa156534b70760d12fe14f26f"},"schema_version":"1.0"},"canonical_sha256":"8e367fc85fd712d94daed163449655dd7ba9c67ea85b8e289fde106f92a2bbed","source":{"kind":"arxiv","id":"1112.1809","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1809","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1809v2","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1809","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"pith_short_12","alias_value":"RY3H7SC724JN","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RY3H7SC724JNSTNO","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RY3H7SC7","created_at":"2026-05-18T12:26:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:RY3H7SC724JNSTNO2FRUJFSV3V","target":"record","payload":{"canonical_record":{"source":{"id":"1112.1809","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-08T11:25:02Z","cross_cats_sorted":[],"title_canon_sha256":"60830ad71f05d973b10cf2392c5a5cdf9807c4d47d9b707d1b25484305175347","abstract_canon_sha256":"15e60e743dc5af72e49b7412850d8fd18757f8baa156534b70760d12fe14f26f"},"schema_version":"1.0"},"canonical_sha256":"8e367fc85fd712d94daed163449655dd7ba9c67ea85b8e289fde106f92a2bbed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:07.422595Z","signature_b64":"bu3hlauhuMPNukIKNWmSYI7QeQmm7Zjb0K7j9cDLugJsDjxPaa+HYQfB8YizPAs4MQc+HPnm+iRyQTJ+/WUGDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e367fc85fd712d94daed163449655dd7ba9c67ea85b8e289fde106f92a2bbed","last_reissued_at":"2026-05-18T04:06:07.422031Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:07.422031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.1809","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:06:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jRAZ1MN6QMw/35P5c6Hp0ZB4K+muMySWPWyM+u129pSlhz439ibYQHG8GyXhrfVbb+hCyCBsDfHX5uPhoHTjAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T21:29:32.423058Z"},"content_sha256":"c9a2b0e44ee3dde5312026e7e4561a2813cbad820f05948b5fa792d83902e3d9","schema_version":"1.0","event_id":"sha256:c9a2b0e44ee3dde5312026e7e4561a2813cbad820f05948b5fa792d83902e3d9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:RY3H7SC724JNSTNO2FRUJFSV3V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quantization of the crossing number of a knot diagram","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Akio Kawauchi, Ayaka Shimizu","submitted_at":"2011-12-08T11:25:02Z","abstract_excerpt":"We introduce the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1809","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":""},"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":"2026-05-18T04:06:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vzbtYVxR/+XnYFTWrhT9VTnhK/vlxzE2S4NaWA52Md7YpSday2scvtSIlfrilxRXyZtqXMsEHDLVS3ChYlpGBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T21:29:32.423529Z"},"content_sha256":"edea4b9c46c41a15f5f6503eb28010ea13a7d2d9f601f9aeda40bee5475c95c8","schema_version":"1.0","event_id":"sha256:edea4b9c46c41a15f5f6503eb28010ea13a7d2d9f601f9aeda40bee5475c95c8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RY3H7SC724JNSTNO2FRUJFSV3V/bundle.json","state_url":"https://pith.science/pith/RY3H7SC724JNSTNO2FRUJFSV3V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RY3H7SC724JNSTNO2FRUJFSV3V/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":"2026-06-25T21:29:32Z","links":{"resolver":"https://pith.science/pith/RY3H7SC724JNSTNO2FRUJFSV3V","bundle":"https://pith.science/pith/RY3H7SC724JNSTNO2FRUJFSV3V/bundle.json","state":"https://pith.science/pith/RY3H7SC724JNSTNO2FRUJFSV3V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RY3H7SC724JNSTNO2FRUJFSV3V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:RY3H7SC724JNSTNO2FRUJFSV3V","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":"15e60e743dc5af72e49b7412850d8fd18757f8baa156534b70760d12fe14f26f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-08T11:25:02Z","title_canon_sha256":"60830ad71f05d973b10cf2392c5a5cdf9807c4d47d9b707d1b25484305175347"},"schema_version":"1.0","source":{"id":"1112.1809","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.1809","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"arxiv_version","alias_value":"1112.1809v2","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.1809","created_at":"2026-05-18T04:06:07Z"},{"alias_kind":"pith_short_12","alias_value":"RY3H7SC724JN","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_16","alias_value":"RY3H7SC724JNSTNO","created_at":"2026-05-18T12:26:41Z"},{"alias_kind":"pith_short_8","alias_value":"RY3H7SC7","created_at":"2026-05-18T12:26:41Z"}],"graph_snapshots":[{"event_id":"sha256:edea4b9c46c41a15f5f6503eb28010ea13a7d2d9f601f9aeda40bee5475c95c8","target":"graph","created_at":"2026-05-18T04:06:07Z","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 the warping crossing polynomial of an oriented knot diagram by using the warping degrees of crossing points of the diagram. Given a closed transversely intersected plane curve, we consider oriented knot diagrams obtained from the plane curve as states to take the sum of the warping crossing polynomials for all the states for the plane curve. As an application, we show that every closed transversely intersected plane curve with even crossing points has two independent canonical orientations and every based closed transversely intersected plane curve with odd crossing points has two","authors_text":"Akio Kawauchi, Ayaka Shimizu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-08T11:25:02Z","title":"Quantization of the crossing number of a knot diagram"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.1809","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:c9a2b0e44ee3dde5312026e7e4561a2813cbad820f05948b5fa792d83902e3d9","target":"record","created_at":"2026-05-18T04:06:07Z","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":"15e60e743dc5af72e49b7412850d8fd18757f8baa156534b70760d12fe14f26f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-12-08T11:25:02Z","title_canon_sha256":"60830ad71f05d973b10cf2392c5a5cdf9807c4d47d9b707d1b25484305175347"},"schema_version":"1.0","source":{"id":"1112.1809","kind":"arxiv","version":2}},"canonical_sha256":"8e367fc85fd712d94daed163449655dd7ba9c67ea85b8e289fde106f92a2bbed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e367fc85fd712d94daed163449655dd7ba9c67ea85b8e289fde106f92a2bbed","first_computed_at":"2026-05-18T04:06:07.422031Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:06:07.422031Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bu3hlauhuMPNukIKNWmSYI7QeQmm7Zjb0K7j9cDLugJsDjxPaa+HYQfB8YizPAs4MQc+HPnm+iRyQTJ+/WUGDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:06:07.422595Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.1809","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9a2b0e44ee3dde5312026e7e4561a2813cbad820f05948b5fa792d83902e3d9","sha256:edea4b9c46c41a15f5f6503eb28010ea13a7d2d9f601f9aeda40bee5475c95c8"],"state_sha256":"10e65220b92c05a98177c20e64806b327f5a1b5f029e56fa16ac7b1e8a25a240"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rGOqi6zprqbTF5MSuQ0p3/0nrZk3H8UrkyptB/JiwkIEji7n5AL0e3QkGtO3BiMLEVVZ7dNuxomJ4SlWHDlvAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T21:29:32.426186Z","bundle_sha256":"b6d7b7e9f5d76f73a8f382b3ea487515392c4d9eb64a9fa139a1f3a88d79c503"}}