{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CQ7XXXIYQIYKJC3ELUOKQON4N3","short_pith_number":"pith:CQ7XXXIY","canonical_record":{"source":{"id":"1608.06600","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2016-08-23T18:46:16Z","cross_cats_sorted":["math.CO","math.GT","math.QA"],"title_canon_sha256":"c1597d4c0f758e93e180a0860400402d650f9a1d642661b7102f94637f8d688c","abstract_canon_sha256":"116ff69e64410dca1dae38b9586127dcd5134f2888a6c7a33f34ecdea8edf2c4"},"schema_version":"1.0"},"canonical_sha256":"143f7bdd188230a48b645d1ca839bc6ed73ac20ad803afd2c55e3cfc1aa6f5b6","source":{"kind":"arxiv","id":"1608.06600","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.06600","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"arxiv_version","alias_value":"1608.06600v1","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06600","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"pith_short_12","alias_value":"CQ7XXXIYQIYK","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CQ7XXXIYQIYKJC3E","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CQ7XXXIY","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CQ7XXXIYQIYKJC3ELUOKQON4N3","target":"record","payload":{"canonical_record":{"source":{"id":"1608.06600","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2016-08-23T18:46:16Z","cross_cats_sorted":["math.CO","math.GT","math.QA"],"title_canon_sha256":"c1597d4c0f758e93e180a0860400402d650f9a1d642661b7102f94637f8d688c","abstract_canon_sha256":"116ff69e64410dca1dae38b9586127dcd5134f2888a6c7a33f34ecdea8edf2c4"},"schema_version":"1.0"},"canonical_sha256":"143f7bdd188230a48b645d1ca839bc6ed73ac20ad803afd2c55e3cfc1aa6f5b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:09.458840Z","signature_b64":"xiTOwABJTgyGmh0kPvGGhcRNaIJ0FsLQCxua1JMgl85X2W4GF/EbUYgTS2+JgRKUv0qea5BdxHHU06ekipq6CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"143f7bdd188230a48b645d1ca839bc6ed73ac20ad803afd2c55e3cfc1aa6f5b6","last_reissued_at":"2026-05-18T00:44:09.458248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:09.458248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.06600","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":"2026-05-18T00:44:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tZPQRQh8ONRJodYG4uVc6Dwe3CAoUMeRbb7+UZPIWYeMko4doVFtX+PJ6xU8petl8R6PfMnU9E/1STVjpvPaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:42:29.675024Z"},"content_sha256":"c34b6c68fe287cae85d6f154fa59f88ad260fd9a97461f70b8905e3db96e1a6c","schema_version":"1.0","event_id":"sha256:c34b6c68fe287cae85d6f154fa59f88ad260fd9a97461f70b8905e3db96e1a6c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CQ7XXXIYQIYKJC3ELUOKQON4N3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"BPS counting for knots and combinatorics on words","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GT","math.QA"],"primary_cat":"hep-th","authors_text":"Piotr Kucharski, Piotr Su{\\l}kowski","submitted_at":"2016-08-23T18:46:16Z","abstract_excerpt":"We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincar\\'e series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06600","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":"2026-05-18T00:44:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bgmbFEvEFrxNvMkEqn3Z5K3+AFOY/3Zo6xJE/8AJRs96lzb+UnKVcJfd8mkLJZ3AVuZwb8Yv8VvWNPRBkSp1AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:42:29.675789Z"},"content_sha256":"0866f6d3364547561e5efdb5243405341b58e4310e2c65d40b17dc8a8aceae0e","schema_version":"1.0","event_id":"sha256:0866f6d3364547561e5efdb5243405341b58e4310e2c65d40b17dc8a8aceae0e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/bundle.json","state_url":"https://pith.science/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/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":"2026-06-11T06:42:29Z","links":{"resolver":"https://pith.science/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3","bundle":"https://pith.science/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/bundle.json","state":"https://pith.science/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CQ7XXXIYQIYKJC3ELUOKQON4N3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CQ7XXXIYQIYKJC3ELUOKQON4N3","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":"116ff69e64410dca1dae38b9586127dcd5134f2888a6c7a33f34ecdea8edf2c4","cross_cats_sorted":["math.CO","math.GT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2016-08-23T18:46:16Z","title_canon_sha256":"c1597d4c0f758e93e180a0860400402d650f9a1d642661b7102f94637f8d688c"},"schema_version":"1.0","source":{"id":"1608.06600","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.06600","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"arxiv_version","alias_value":"1608.06600v1","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.06600","created_at":"2026-05-18T00:44:09Z"},{"alias_kind":"pith_short_12","alias_value":"CQ7XXXIYQIYK","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CQ7XXXIYQIYKJC3E","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CQ7XXXIY","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:0866f6d3364547561e5efdb5243405341b58e4310e2c65d40b17dc8a8aceae0e","target":"graph","created_at":"2026-05-18T00:44:09Z","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 discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincar\\'e series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to th","authors_text":"Piotr Kucharski, Piotr Su{\\l}kowski","cross_cats":["math.CO","math.GT","math.QA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2016-08-23T18:46:16Z","title":"BPS counting for knots and combinatorics on words"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06600","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:c34b6c68fe287cae85d6f154fa59f88ad260fd9a97461f70b8905e3db96e1a6c","target":"record","created_at":"2026-05-18T00:44:09Z","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":"116ff69e64410dca1dae38b9586127dcd5134f2888a6c7a33f34ecdea8edf2c4","cross_cats_sorted":["math.CO","math.GT","math.QA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2016-08-23T18:46:16Z","title_canon_sha256":"c1597d4c0f758e93e180a0860400402d650f9a1d642661b7102f94637f8d688c"},"schema_version":"1.0","source":{"id":"1608.06600","kind":"arxiv","version":1}},"canonical_sha256":"143f7bdd188230a48b645d1ca839bc6ed73ac20ad803afd2c55e3cfc1aa6f5b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"143f7bdd188230a48b645d1ca839bc6ed73ac20ad803afd2c55e3cfc1aa6f5b6","first_computed_at":"2026-05-18T00:44:09.458248Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:09.458248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xiTOwABJTgyGmh0kPvGGhcRNaIJ0FsLQCxua1JMgl85X2W4GF/EbUYgTS2+JgRKUv0qea5BdxHHU06ekipq6CA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:09.458840Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.06600","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c34b6c68fe287cae85d6f154fa59f88ad260fd9a97461f70b8905e3db96e1a6c","sha256:0866f6d3364547561e5efdb5243405341b58e4310e2c65d40b17dc8a8aceae0e"],"state_sha256":"a0deb03614798766c97c7566999726d8fd962b29c89a95f295398eaff42a45d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2XaQvzImrATuuEypcS7QnHBaCSoiWuXS3vIE9QzBuzhiv16ybhS4Iabd7unl5ss3HvwsVu7ckbts6Pa5jO50BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:42:29.715167Z","bundle_sha256":"f8f605f3a0a32121d601dc75adc9a871b216ded9ca32ff6cbd8b1e50e6038227"}}