{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:QCBB6XWFUNZ4ZKBQCXCNK655DB","short_pith_number":"pith:QCBB6XWF","canonical_record":{"source":{"id":"1106.3784","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-19T22:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"8ac516fd1cb54f0aae2e2d0fc3f529370ce3ffef132dfe05e36caa8cb75ed61c","abstract_canon_sha256":"59de49f75b4c0b207bb6ba0bc4240e21e60320cd03cd5ddc06e8af2d495ab9f7"},"schema_version":"1.0"},"canonical_sha256":"80821f5ec5a373cca83015c4d57bbd186fe0090950b289e9dbadae669c4909c8","source":{"kind":"arxiv","id":"1106.3784","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.3784","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"arxiv_version","alias_value":"1106.3784v2","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3784","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"pith_short_12","alias_value":"QCBB6XWFUNZ4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QCBB6XWFUNZ4ZKBQ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QCBB6XWF","created_at":"2026-05-18T12:26:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:QCBB6XWFUNZ4ZKBQCXCNK655DB","target":"record","payload":{"canonical_record":{"source":{"id":"1106.3784","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-19T22:02:20Z","cross_cats_sorted":[],"title_canon_sha256":"8ac516fd1cb54f0aae2e2d0fc3f529370ce3ffef132dfe05e36caa8cb75ed61c","abstract_canon_sha256":"59de49f75b4c0b207bb6ba0bc4240e21e60320cd03cd5ddc06e8af2d495ab9f7"},"schema_version":"1.0"},"canonical_sha256":"80821f5ec5a373cca83015c4d57bbd186fe0090950b289e9dbadae669c4909c8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:38.603452Z","signature_b64":"m+T2IInSvBxTIl3S5VRgRRw+hnPk0rqXQP7ArXmK9i+Kit9OKwsKisWsZ2IiciUngygrO/i0JFH/iwEo0sS8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80821f5ec5a373cca83015c4d57bbd186fe0090950b289e9dbadae669c4909c8","last_reissued_at":"2026-05-18T04:05:38.602877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:38.602877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.3784","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":"2026-05-18T04:05:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X0chSQEoQujoShUqCJIKx4sptcas9L5c5S7V/tmYxDHVS5kXg7PIMD2uGsr3arHKz1S5K5oSk/br3rjWuls4Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T11:03:57.610031Z"},"content_sha256":"ef0f3889d9e7357509adfcf5aa09910b8a16037a11275764d5df6c642ffdd001","schema_version":"1.0","event_id":"sha256:ef0f3889d9e7357509adfcf5aa09910b8a16037a11275764d5df6c642ffdd001"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:QCBB6XWFUNZ4ZKBQCXCNK655DB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mirror-Curves and Knot Mosaics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Ana Zekovic, Ljiljana Radovic, Radmila Sazdanovic, Slavik V. Jablan","submitted_at":"2011-06-19T22:02:20Z","abstract_excerpt":"Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams [3,7,22,24]. Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3784","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":"2026-05-18T04:05:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sd2y5So9gdxWDSdOYorDcM+dBpfxNtxBqE+V7tP2gyvHMJlCdFmn9DJvxok0euTxuK3p8fimEA5JRwCOSa3ZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T11:03:57.610398Z"},"content_sha256":"1eb64216dbf673774dfcbb2f4d3563b3a626f21660de464e230ebbc2e033f4b1","schema_version":"1.0","event_id":"sha256:1eb64216dbf673774dfcbb2f4d3563b3a626f21660de464e230ebbc2e033f4b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/bundle.json","state_url":"https://pith.science/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/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":"2026-06-26T11:03:57Z","links":{"resolver":"https://pith.science/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB","bundle":"https://pith.science/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/bundle.json","state":"https://pith.science/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QCBB6XWFUNZ4ZKBQCXCNK655DB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:QCBB6XWFUNZ4ZKBQCXCNK655DB","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":"59de49f75b4c0b207bb6ba0bc4240e21e60320cd03cd5ddc06e8af2d495ab9f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-19T22:02:20Z","title_canon_sha256":"8ac516fd1cb54f0aae2e2d0fc3f529370ce3ffef132dfe05e36caa8cb75ed61c"},"schema_version":"1.0","source":{"id":"1106.3784","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.3784","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"arxiv_version","alias_value":"1106.3784v2","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.3784","created_at":"2026-05-18T04:05:38Z"},{"alias_kind":"pith_short_12","alias_value":"QCBB6XWFUNZ4","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"QCBB6XWFUNZ4ZKBQ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"QCBB6XWF","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:1eb64216dbf673774dfcbb2f4d3563b3a626f21660de464e230ebbc2e033f4b1","target":"graph","created_at":"2026-05-18T04:05:38Z","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":"Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology [2,3,7,24], we construct the more concise representation of knot mosaics and grid diagrams via mirror-curves. Tame knot theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams [3,7,22,24]. Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links ","authors_text":"Ana Zekovic, Ljiljana Radovic, Radmila Sazdanovic, Slavik V. Jablan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-19T22:02:20Z","title":"Mirror-Curves and Knot Mosaics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.3784","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:ef0f3889d9e7357509adfcf5aa09910b8a16037a11275764d5df6c642ffdd001","target":"record","created_at":"2026-05-18T04:05:38Z","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":"59de49f75b4c0b207bb6ba0bc4240e21e60320cd03cd5ddc06e8af2d495ab9f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-06-19T22:02:20Z","title_canon_sha256":"8ac516fd1cb54f0aae2e2d0fc3f529370ce3ffef132dfe05e36caa8cb75ed61c"},"schema_version":"1.0","source":{"id":"1106.3784","kind":"arxiv","version":2}},"canonical_sha256":"80821f5ec5a373cca83015c4d57bbd186fe0090950b289e9dbadae669c4909c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"80821f5ec5a373cca83015c4d57bbd186fe0090950b289e9dbadae669c4909c8","first_computed_at":"2026-05-18T04:05:38.602877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:38.602877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m+T2IInSvBxTIl3S5VRgRRw+hnPk0rqXQP7ArXmK9i+Kit9OKwsKisWsZ2IiciUngygrO/i0JFH/iwEo0sS8Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:38.603452Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.3784","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ef0f3889d9e7357509adfcf5aa09910b8a16037a11275764d5df6c642ffdd001","sha256:1eb64216dbf673774dfcbb2f4d3563b3a626f21660de464e230ebbc2e033f4b1"],"state_sha256":"86db9230a686ae72f3432fe2339a6e4bdaaabe8b48834f340554eebe29f50c64"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"S0YtZIryDMJBd8wlepHFcbxL2Fsq9rigaeyHqMgLURsLa/7YB3Z0+5vzxzKc7q9MVO8rNxwo1S3AZT1LdbY5Bw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T11:03:57.612349Z","bundle_sha256":"590f099e1316398573db9bd3a02d4903e19b4bce7197707938c098bd79d7a4f5"}}