{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:YDGYPK3FWYXRWF4B27AB6A3L4J","short_pith_number":"pith:YDGYPK3F","canonical_record":{"source":{"id":"1707.04948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-07-16T21:35:23Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"9899d639381a5e2b7015f7c2c401a6d26c7fa87e4332d6fd198e4becbbc615c8","abstract_canon_sha256":"473d32b8af4cd717221ac770e9e01283e159a808772fb5a3bfb9bcf3e530fdba"},"schema_version":"1.0"},"canonical_sha256":"c0cd87ab65b62f1b1781d7c01f036be27b25785177106ebf572feb5f22b32cd9","source":{"kind":"arxiv","id":"1707.04948","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04948","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04948v1","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04948","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"pith_short_12","alias_value":"YDGYPK3FWYXR","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YDGYPK3FWYXRWF4B","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YDGYPK3F","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:YDGYPK3FWYXRWF4B27AB6A3L4J","target":"record","payload":{"canonical_record":{"source":{"id":"1707.04948","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-07-16T21:35:23Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"9899d639381a5e2b7015f7c2c401a6d26c7fa87e4332d6fd198e4becbbc615c8","abstract_canon_sha256":"473d32b8af4cd717221ac770e9e01283e159a808772fb5a3bfb9bcf3e530fdba"},"schema_version":"1.0"},"canonical_sha256":"c0cd87ab65b62f1b1781d7c01f036be27b25785177106ebf572feb5f22b32cd9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:11.731111Z","signature_b64":"d0ULmhCDPdJjr4cjC2x+Ur1m0Z4rpP2rfP4bwbrKgpIjYB4VLMSxs03n85H77jXZBuyNcTeUdwyXU1X+xtnVAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0cd87ab65b62f1b1781d7c01f036be27b25785177106ebf572feb5f22b32cd9","last_reissued_at":"2026-05-18T00:40:11.730566Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:11.730566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.04948","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":"2026-05-18T00:40:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nkABesiMf2/bx8xqzQCn0AE9Qu4Ho7zc8NMHgl6lz1Wo0F/7H9uMLpBVhKKxcmLBzLf/lBrFjHpNXXaupFttCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T10:26:30.256597Z"},"content_sha256":"082a6bc13f6ce8af4c72d06362e845453bf14eec0560a6c9be19bad478803c6b","schema_version":"1.0","event_id":"sha256:082a6bc13f6ce8af4c72d06362e845453bf14eec0560a6c9be19bad478803c6b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:YDGYPK3FWYXRWF4B27AB6A3L4J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ruling polynomials and augmentations for Legendrian tangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.SG","authors_text":"Tao Su","submitted_at":"2017-07-16T21:35:23Z","abstract_excerpt":"Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links (up to a normalized factor) \\cite{HR15}. In this article, we generalize this picture to Legendrian tangles, which are morally the pieces obtained by cutting Legendrian link fronts along 2 vertical lines. Moreover, we show that the Ruling polynomials for Legendrian tangles satisfy the composition axiom. In the special case of Legendrian knots, our arguments pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04948","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":"2026-05-18T00:40:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N6XfzHqS763u1fMv13PHL6Rn37zzZycYck3JQMREGqoZFT7YlRRrF3EX3q7qL8XGSb8j4ndJT5d6Xkx48jCbCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T10:26:30.256938Z"},"content_sha256":"314a1e3732fb3e9d7282fb3725641c5c44ffc48ed366a7411b16ca5c2715d01e","schema_version":"1.0","event_id":"sha256:314a1e3732fb3e9d7282fb3725641c5c44ffc48ed366a7411b16ca5c2715d01e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/bundle.json","state_url":"https://pith.science/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/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":"2026-06-20T10:26:30Z","links":{"resolver":"https://pith.science/pith/YDGYPK3FWYXRWF4B27AB6A3L4J","bundle":"https://pith.science/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/bundle.json","state":"https://pith.science/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YDGYPK3FWYXRWF4B27AB6A3L4J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YDGYPK3FWYXRWF4B27AB6A3L4J","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":"473d32b8af4cd717221ac770e9e01283e159a808772fb5a3bfb9bcf3e530fdba","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-07-16T21:35:23Z","title_canon_sha256":"9899d639381a5e2b7015f7c2c401a6d26c7fa87e4332d6fd198e4becbbc615c8"},"schema_version":"1.0","source":{"id":"1707.04948","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04948","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04948v1","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04948","created_at":"2026-05-18T00:40:11Z"},{"alias_kind":"pith_short_12","alias_value":"YDGYPK3FWYXR","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YDGYPK3FWYXRWF4B","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YDGYPK3F","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:314a1e3732fb3e9d7282fb3725641c5c44ffc48ed366a7411b16ca5c2715d01e","target":"graph","created_at":"2026-05-18T00:40:11Z","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":"Associated to Legendrian links in the standard contact three-space, Ruling polynomials are Legendrian isotopy invariants, which also compute augmentation numbers, that is, the points-counting of augmentation varieties for Legendrian links (up to a normalized factor) \\cite{HR15}. In this article, we generalize this picture to Legendrian tangles, which are morally the pieces obtained by cutting Legendrian link fronts along 2 vertical lines. Moreover, we show that the Ruling polynomials for Legendrian tangles satisfy the composition axiom. In the special case of Legendrian knots, our arguments pr","authors_text":"Tao Su","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-07-16T21:35:23Z","title":"Ruling polynomials and augmentations for Legendrian tangles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04948","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":"sha256:082a6bc13f6ce8af4c72d06362e845453bf14eec0560a6c9be19bad478803c6b","target":"record","created_at":"2026-05-18T00:40:11Z","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":"473d32b8af4cd717221ac770e9e01283e159a808772fb5a3bfb9bcf3e530fdba","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2017-07-16T21:35:23Z","title_canon_sha256":"9899d639381a5e2b7015f7c2c401a6d26c7fa87e4332d6fd198e4becbbc615c8"},"schema_version":"1.0","source":{"id":"1707.04948","kind":"arxiv","version":1}},"canonical_sha256":"c0cd87ab65b62f1b1781d7c01f036be27b25785177106ebf572feb5f22b32cd9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0cd87ab65b62f1b1781d7c01f036be27b25785177106ebf572feb5f22b32cd9","first_computed_at":"2026-05-18T00:40:11.730566Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:11.730566Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d0ULmhCDPdJjr4cjC2x+Ur1m0Z4rpP2rfP4bwbrKgpIjYB4VLMSxs03n85H77jXZBuyNcTeUdwyXU1X+xtnVAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:11.731111Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.04948","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:082a6bc13f6ce8af4c72d06362e845453bf14eec0560a6c9be19bad478803c6b","sha256:314a1e3732fb3e9d7282fb3725641c5c44ffc48ed366a7411b16ca5c2715d01e"],"state_sha256":"472d682e4df239fe4e87e44f1310442bee870c557e51d2980670d9d2a84a8e2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"10klbO5KmBF8uIGuBq/d11BEfntQCsuy2P+mg0g47cz+UX3+dcRdkRgipTeewKk4EV1WtHavo23dKxYXkmp7DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T10:26:30.258815Z","bundle_sha256":"7568a7c0e3062a7cd073a61eb08d6d8f69d2ef7d2de4b2e024803b0e4251aea9"}}