{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:45LP76RZPQGQXTPDKUM4CZUPMK","short_pith_number":"pith:45LP76RZ","canonical_record":{"source":{"id":"1209.0048","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-01T02:40:40Z","cross_cats_sorted":[],"title_canon_sha256":"d733993c05a4ba99fc6827d438b53386cd0bff345334a7bf49ec2666519ff927","abstract_canon_sha256":"e4f88f095b8faab9c85e616ae6bffe6401e8db48cc6228d74b895318da1506bb"},"schema_version":"1.0"},"canonical_sha256":"e756fffa397c0d0bcde35519c1668f62bd1bd3803ecba9d376438c02b6d26da4","source":{"kind":"arxiv","id":"1209.0048","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.0048","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"arxiv_version","alias_value":"1209.0048v1","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0048","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"pith_short_12","alias_value":"45LP76RZPQGQ","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"45LP76RZPQGQXTPD","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"45LP76RZ","created_at":"2026-05-18T12:26:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:45LP76RZPQGQXTPDKUM4CZUPMK","target":"record","payload":{"canonical_record":{"source":{"id":"1209.0048","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-01T02:40:40Z","cross_cats_sorted":[],"title_canon_sha256":"d733993c05a4ba99fc6827d438b53386cd0bff345334a7bf49ec2666519ff927","abstract_canon_sha256":"e4f88f095b8faab9c85e616ae6bffe6401e8db48cc6228d74b895318da1506bb"},"schema_version":"1.0"},"canonical_sha256":"e756fffa397c0d0bcde35519c1668f62bd1bd3803ecba9d376438c02b6d26da4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:30.285337Z","signature_b64":"xZ3H2+tuhjovuIAqDb8K4D63CLQSyEZPKCvxWxi2XTW79L/NkPKFnc2zepwNCZrGSUPyApp6lq5Q1mBVP7CrBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e756fffa397c0d0bcde35519c1668f62bd1bd3803ecba9d376438c02b6d26da4","last_reissued_at":"2026-05-18T00:44:30.284724Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:30.284724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.0048","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":"2026-05-18T00:44:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YbOaveOlIUMsJUhhdgfwLd4/e7HnKhG+IaeD/RCde9r7ziLDND98cfwQ0PBe+GI1lUOMjRiQJTa81IWYrNuwCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:42:20.048657Z"},"content_sha256":"432c78bc338a68d2d662484ff9bc9b0f32238578b30167ae9cf3a320360a3a30","schema_version":"1.0","event_id":"sha256:432c78bc338a68d2d662484ff9bc9b0f32238578b30167ae9cf3a320360a3a30"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:45LP76RZPQGQXTPDKUM4CZUPMK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Upper bound on lattice stick number of knots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Kyungpyo Hong, Seungsang Oh, Sungjong No","submitted_at":"2012-09-01T02:40:40Z","abstract_excerpt":"The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot $K$, except trefoil knot, in terms of the minimal crossing number $c(K)$ which is $s_L(K) \\leq 3 c(K) +2$. Moreover if $K$ is a non-alternating prime knot, then $s_L(K) \\leq 3 c(K) - 4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0048","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":"2026-05-18T00:44:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p7c7kPoLnCC1F3LRIrA9I/j0VpfEUJqhQaePXi7r3acwUXMU7e9xxGodJLXSLvGa7xee83H/6mo7vnqweyoRBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-20T17:42:20.049005Z"},"content_sha256":"ada5b75ceab87556e6d30b1f9d0395ccff76d522dfc352cd802b35cb5f0399c3","schema_version":"1.0","event_id":"sha256:ada5b75ceab87556e6d30b1f9d0395ccff76d522dfc352cd802b35cb5f0399c3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/45LP76RZPQGQXTPDKUM4CZUPMK/bundle.json","state_url":"https://pith.science/pith/45LP76RZPQGQXTPDKUM4CZUPMK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/45LP76RZPQGQXTPDKUM4CZUPMK/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":"2026-05-20T17:42:20Z","links":{"resolver":"https://pith.science/pith/45LP76RZPQGQXTPDKUM4CZUPMK","bundle":"https://pith.science/pith/45LP76RZPQGQXTPDKUM4CZUPMK/bundle.json","state":"https://pith.science/pith/45LP76RZPQGQXTPDKUM4CZUPMK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/45LP76RZPQGQXTPDKUM4CZUPMK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:45LP76RZPQGQXTPDKUM4CZUPMK","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":"e4f88f095b8faab9c85e616ae6bffe6401e8db48cc6228d74b895318da1506bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-01T02:40:40Z","title_canon_sha256":"d733993c05a4ba99fc6827d438b53386cd0bff345334a7bf49ec2666519ff927"},"schema_version":"1.0","source":{"id":"1209.0048","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.0048","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"arxiv_version","alias_value":"1209.0048v1","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.0048","created_at":"2026-05-18T00:44:30Z"},{"alias_kind":"pith_short_12","alias_value":"45LP76RZPQGQ","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_16","alias_value":"45LP76RZPQGQXTPD","created_at":"2026-05-18T12:26:53Z"},{"alias_kind":"pith_short_8","alias_value":"45LP76RZ","created_at":"2026-05-18T12:26:53Z"}],"graph_snapshots":[{"event_id":"sha256:ada5b75ceab87556e6d30b1f9d0395ccff76d522dfc352cd802b35cb5f0399c3","target":"graph","created_at":"2026-05-18T00:44:30Z","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":"The lattice stick number $s_L(K)$ of a knot $K$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $K$ in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot $K$, except trefoil knot, in terms of the minimal crossing number $c(K)$ which is $s_L(K) \\leq 3 c(K) +2$. Moreover if $K$ is a non-alternating prime knot, then $s_L(K) \\leq 3 c(K) - 4$.","authors_text":"Kyungpyo Hong, Seungsang Oh, Sungjong No","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-01T02:40:40Z","title":"Upper bound on lattice stick number of knots"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0048","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":"sha256:432c78bc338a68d2d662484ff9bc9b0f32238578b30167ae9cf3a320360a3a30","target":"record","created_at":"2026-05-18T00:44:30Z","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":"e4f88f095b8faab9c85e616ae6bffe6401e8db48cc6228d74b895318da1506bb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-01T02:40:40Z","title_canon_sha256":"d733993c05a4ba99fc6827d438b53386cd0bff345334a7bf49ec2666519ff927"},"schema_version":"1.0","source":{"id":"1209.0048","kind":"arxiv","version":1}},"canonical_sha256":"e756fffa397c0d0bcde35519c1668f62bd1bd3803ecba9d376438c02b6d26da4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e756fffa397c0d0bcde35519c1668f62bd1bd3803ecba9d376438c02b6d26da4","first_computed_at":"2026-05-18T00:44:30.284724Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:30.284724Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xZ3H2+tuhjovuIAqDb8K4D63CLQSyEZPKCvxWxi2XTW79L/NkPKFnc2zepwNCZrGSUPyApp6lq5Q1mBVP7CrBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:30.285337Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.0048","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:432c78bc338a68d2d662484ff9bc9b0f32238578b30167ae9cf3a320360a3a30","sha256:ada5b75ceab87556e6d30b1f9d0395ccff76d522dfc352cd802b35cb5f0399c3"],"state_sha256":"fbbb481953e7e8c8882f5adde30d112982dae420333e4b2e92a499554fcebb78"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bCkUGCnheeqL2a596Cqc2eEsyJxi2VwwmqTS2QIBgV1Y2+UljfH5njSr23GbeARAlr2UyA0RFHYVjwpG2mW9Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-20T17:42:20.051028Z","bundle_sha256":"9cb1602dbfe53e5d1e86dfd2d464a705e453f48cb86ec08e14639d8c784dad88"}}