{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:SFGPVFGDGYJPSSDEVYTBZH2HS2","short_pith_number":"pith:SFGPVFGD","canonical_record":{"source":{"id":"1501.02421","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GT","submitted_at":"2015-01-11T06:26:33Z","cross_cats_sorted":[],"title_canon_sha256":"81051602be933bc0d0cc39bd35731a04dc3f233a62dddd0fa045d8355c1f512d","abstract_canon_sha256":"a0de69b7f3e2bf124ab317219a7cd8eefec606767c4275211abab8297737e4a9"},"schema_version":"1.0"},"canonical_sha256":"914cfa94c33612f94864ae261c9f4796af40aa5caaf1b8e3f032bd75962fe786","source":{"kind":"arxiv","id":"1501.02421","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02421","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02421v1","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02421","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"SFGPVFGDGYJP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"SFGPVFGDGYJPSSDE","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"SFGPVFGD","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:SFGPVFGDGYJPSSDEVYTBZH2HS2","target":"record","payload":{"canonical_record":{"source":{"id":"1501.02421","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GT","submitted_at":"2015-01-11T06:26:33Z","cross_cats_sorted":[],"title_canon_sha256":"81051602be933bc0d0cc39bd35731a04dc3f233a62dddd0fa045d8355c1f512d","abstract_canon_sha256":"a0de69b7f3e2bf124ab317219a7cd8eefec606767c4275211abab8297737e4a9"},"schema_version":"1.0"},"canonical_sha256":"914cfa94c33612f94864ae261c9f4796af40aa5caaf1b8e3f032bd75962fe786","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:29:38.003166Z","signature_b64":"S1dZAK8WaI1oQ8V5lyzRwCvvogq/4M7Jqdtoigq8QcNGE3UG4BXitt4RkJKzDPAiYsq/EAn4DRC9sj3JQN5rAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"914cfa94c33612f94864ae261c9f4796af40aa5caaf1b8e3f032bd75962fe786","last_reissued_at":"2026-05-18T02:29:38.002805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:29:38.002805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.02421","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j+r2ga6l77bMnvlkFVbu/Uv5Vuk98j8UmVuw21u9pF/PZ39TBEag5la4aMSk2nOLtmN6gxMP35585VMWTc1iAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:56:35.712128Z"},"content_sha256":"214a9e4808c4ce3e872d97951afef13fd46f55d1ffe32652ef3073e9da91f4f8","schema_version":"1.0","event_id":"sha256:214a9e4808c4ce3e872d97951afef13fd46f55d1ffe32652ef3073e9da91f4f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:SFGPVFGDGYJPSSDEVYTBZH2HS2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal sufficient sets of colors and minimum number of colors","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jun Ge, Lianzhu Zhang, Louis H. Kauffman, Pedro Lopes, Xian'an Jin","submitted_at":"2015-01-11T06:26:33Z","abstract_excerpt":"In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \\geq 5 for p \\geq 11 and mincol_p L \\geq 6 for p \\geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02421","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:29:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xxf37fLxSZGwDtG89k7DbTN1vN8wFkx7mug3uvstevcPj+Us1Luq8FlyrNSpfHchrgIejLZc+zMjaXtQ54iyBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T23:56:35.712499Z"},"content_sha256":"cdde7cc85668d8d57bfd9474ac35a959fdb352b9463075f72dbede6e3aedc3a6","schema_version":"1.0","event_id":"sha256:cdde7cc85668d8d57bfd9474ac35a959fdb352b9463075f72dbede6e3aedc3a6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/bundle.json","state_url":"https://pith.science/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/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":"2026-06-25T23:56:35Z","links":{"resolver":"https://pith.science/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2","bundle":"https://pith.science/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/bundle.json","state":"https://pith.science/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SFGPVFGDGYJPSSDEVYTBZH2HS2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SFGPVFGDGYJPSSDEVYTBZH2HS2","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":"a0de69b7f3e2bf124ab317219a7cd8eefec606767c4275211abab8297737e4a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GT","submitted_at":"2015-01-11T06:26:33Z","title_canon_sha256":"81051602be933bc0d0cc39bd35731a04dc3f233a62dddd0fa045d8355c1f512d"},"schema_version":"1.0","source":{"id":"1501.02421","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.02421","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"arxiv_version","alias_value":"1501.02421v1","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.02421","created_at":"2026-05-18T02:29:38Z"},{"alias_kind":"pith_short_12","alias_value":"SFGPVFGDGYJP","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"SFGPVFGDGYJPSSDE","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"SFGPVFGD","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:cdde7cc85668d8d57bfd9474ac35a959fdb352b9463075f72dbede6e3aedc3a6","target":"graph","created_at":"2026-05-18T02:29:38Z","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":"In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \\geq 5 for p \\geq 11 and mincol_p L \\geq 6 for p \\geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. ","authors_text":"Jun Ge, Lianzhu Zhang, Louis H. Kauffman, Pedro Lopes, Xian'an Jin","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GT","submitted_at":"2015-01-11T06:26:33Z","title":"Minimal sufficient sets of colors and minimum number of colors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02421","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":"sha256:214a9e4808c4ce3e872d97951afef13fd46f55d1ffe32652ef3073e9da91f4f8","target":"record","created_at":"2026-05-18T02:29:38Z","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":"a0de69b7f3e2bf124ab317219a7cd8eefec606767c4275211abab8297737e4a9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GT","submitted_at":"2015-01-11T06:26:33Z","title_canon_sha256":"81051602be933bc0d0cc39bd35731a04dc3f233a62dddd0fa045d8355c1f512d"},"schema_version":"1.0","source":{"id":"1501.02421","kind":"arxiv","version":1}},"canonical_sha256":"914cfa94c33612f94864ae261c9f4796af40aa5caaf1b8e3f032bd75962fe786","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"914cfa94c33612f94864ae261c9f4796af40aa5caaf1b8e3f032bd75962fe786","first_computed_at":"2026-05-18T02:29:38.002805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:38.002805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S1dZAK8WaI1oQ8V5lyzRwCvvogq/4M7Jqdtoigq8QcNGE3UG4BXitt4RkJKzDPAiYsq/EAn4DRC9sj3JQN5rAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:38.003166Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.02421","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:214a9e4808c4ce3e872d97951afef13fd46f55d1ffe32652ef3073e9da91f4f8","sha256:cdde7cc85668d8d57bfd9474ac35a959fdb352b9463075f72dbede6e3aedc3a6"],"state_sha256":"412a86600ece14b9a7556cd1a40ac3cf6506aaafb2de9badce603a6f139295e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jb8bTwZU3d9+dQjZX7fXCR/hPbnK7l3n9rhY7EB41cPX5/bY4R1xn2HuSXL0bRGiEEC8OoXE9w+/nEvbRn3YAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T23:56:35.714585Z","bundle_sha256":"6da86bcc3c4989477157c53500912b539d3645fb7f2aad9e4d3bd20abcd47a05"}}