{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:R44ZQDA4CXOCJGJBTHTLPMECL2","short_pith_number":"pith:R44ZQDA4","canonical_record":{"source":{"id":"math/0512088","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2005-12-04T21:27:20Z","cross_cats_sorted":[],"title_canon_sha256":"304d079656c8f0e37f0a0303d11a0a487d44c091f652ce3835e0201c442c45d2","abstract_canon_sha256":"2703afaf4d75cb5e0a5991d7cdab0546522b08b341128c58136191a0d208678b"},"schema_version":"1.0"},"canonical_sha256":"8f39980c1c15dc24992199e6b7b0825e8b7e8c79f7bccf1b7282f91b48055c45","source":{"kind":"arxiv","id":"math/0512088","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512088","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512088v4","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512088","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"pith_short_12","alias_value":"R44ZQDA4CXOC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"R44ZQDA4CXOCJGJB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"R44ZQDA4","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:R44ZQDA4CXOCJGJBTHTLPMECL2","target":"record","payload":{"canonical_record":{"source":{"id":"math/0512088","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.GT","submitted_at":"2005-12-04T21:27:20Z","cross_cats_sorted":[],"title_canon_sha256":"304d079656c8f0e37f0a0303d11a0a487d44c091f652ce3835e0201c442c45d2","abstract_canon_sha256":"2703afaf4d75cb5e0a5991d7cdab0546522b08b341128c58136191a0d208678b"},"schema_version":"1.0"},"canonical_sha256":"8f39980c1c15dc24992199e6b7b0825e8b7e8c79f7bccf1b7282f91b48055c45","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:35.610435Z","signature_b64":"QxBDD80rhLZfCjroLCxgmzeHc8gTU4z/8dvF4CTj5uOryA3r6UNhUhxv+x/eS2f8T/whX0UmfSqXpeaRwCXeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f39980c1c15dc24992199e6b7b0825e8b7e8c79f7bccf1b7282f91b48055c45","last_reissued_at":"2026-05-18T04:08:35.609699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:35.609699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0512088","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:08:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GHVK9dHGuZKxXgXDXbe7pMBzqp7Bv7Vk8H1vXf4JLf/kwSi0oxuxesbckud+3luF8+rSLU6rsdKL2Gdwo0kJDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:08:18.282640Z"},"content_sha256":"b258c33fe6662189b8b2817aedf331b422efa90c807e91e8c166772dfc4a4d0e","schema_version":"1.0","event_id":"sha256:b258c33fe6662189b8b2817aedf331b422efa90c807e91e8c166772dfc4a4d0e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:R44ZQDA4CXOCJGJBTHTLPMECL2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the minimum number of colors for knots","license":"","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Louis H. Kauffman, Pedro Lopes","submitted_at":"2005-12-04T21:27:20Z","abstract_excerpt":"In this article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva Transformations. Each of these Transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512088","kind":"arxiv","version":4},"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":"2026-05-18T04:08:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wsdDLPu95BSIVQ+facJzlnKSvke5QYXnc/Tkq3M+OGjIiW/eDHPIFAogOld7TFAf1e4WWeHvAZ3/9X0dL3OZDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T15:08:18.282985Z"},"content_sha256":"c69901e2084f81e3823ab47252cf8860a2e2d204c11374d987718b1f0c42416f","schema_version":"1.0","event_id":"sha256:c69901e2084f81e3823ab47252cf8860a2e2d204c11374d987718b1f0c42416f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/bundle.json","state_url":"https://pith.science/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/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":"2026-06-09T15:08:18Z","links":{"resolver":"https://pith.science/pith/R44ZQDA4CXOCJGJBTHTLPMECL2","bundle":"https://pith.science/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/bundle.json","state":"https://pith.science/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/R44ZQDA4CXOCJGJBTHTLPMECL2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:R44ZQDA4CXOCJGJBTHTLPMECL2","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":"2703afaf4d75cb5e0a5991d7cdab0546522b08b341128c58136191a0d208678b","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2005-12-04T21:27:20Z","title_canon_sha256":"304d079656c8f0e37f0a0303d11a0a487d44c091f652ce3835e0201c442c45d2"},"schema_version":"1.0","source":{"id":"math/0512088","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0512088","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"arxiv_version","alias_value":"math/0512088v4","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0512088","created_at":"2026-05-18T04:08:35Z"},{"alias_kind":"pith_short_12","alias_value":"R44ZQDA4CXOC","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"R44ZQDA4CXOCJGJB","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"R44ZQDA4","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:c69901e2084f81e3823ab47252cf8860a2e2d204c11374d987718b1f0c42416f","target":"graph","created_at":"2026-05-18T04:08:35Z","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 article we take up the calculation of the minimum number of colors needed to produce a non-trivial coloring of a knot. This is a knot invariant and we use the torus knots of type (2, n) as our case study. We calculate the minima in some cases. In other cases we estimate upper bounds for these minima leaning on the features of modular arithmetic. We introduce a sequence of transformations on colored diagrams called Teneva Transformations. Each of these Transformations reduces the number of colors in the diagrams by one (up to a point). This allows us to further decrease the upper bounds","authors_text":"Louis H. Kauffman, Pedro Lopes","cross_cats":[],"headline":"","license":"","primary_cat":"math.GT","submitted_at":"2005-12-04T21:27:20Z","title":"On the minimum number of colors for knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0512088","kind":"arxiv","version":4},"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:b258c33fe6662189b8b2817aedf331b422efa90c807e91e8c166772dfc4a4d0e","target":"record","created_at":"2026-05-18T04:08:35Z","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":"2703afaf4d75cb5e0a5991d7cdab0546522b08b341128c58136191a0d208678b","cross_cats_sorted":[],"license":"","primary_cat":"math.GT","submitted_at":"2005-12-04T21:27:20Z","title_canon_sha256":"304d079656c8f0e37f0a0303d11a0a487d44c091f652ce3835e0201c442c45d2"},"schema_version":"1.0","source":{"id":"math/0512088","kind":"arxiv","version":4}},"canonical_sha256":"8f39980c1c15dc24992199e6b7b0825e8b7e8c79f7bccf1b7282f91b48055c45","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8f39980c1c15dc24992199e6b7b0825e8b7e8c79f7bccf1b7282f91b48055c45","first_computed_at":"2026-05-18T04:08:35.609699Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:08:35.609699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QxBDD80rhLZfCjroLCxgmzeHc8gTU4z/8dvF4CTj5uOryA3r6UNhUhxv+x/eS2f8T/whX0UmfSqXpeaRwCXeDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:08:35.610435Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0512088","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b258c33fe6662189b8b2817aedf331b422efa90c807e91e8c166772dfc4a4d0e","sha256:c69901e2084f81e3823ab47252cf8860a2e2d204c11374d987718b1f0c42416f"],"state_sha256":"59eb40e2020c161b15a0c659172b1827bbc7b27eb5f54490e38b0df95c469c79"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5AUINDJfzvBwrHD8ol5CMAQ23ttjOqhA1F8EK2JEmCn6jVdL7mVHdIJjKMtZyQ7bprCI1ByEqmJslHUgmgctCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T15:08:18.284943Z","bundle_sha256":"bcd2dc8957b34cca2e8349bff9496be31b20842c5987f37d489d03fdc89c86cd"}}