{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:W6AYHFA6TJDL2H7TDIPBESOX5R","short_pith_number":"pith:W6AYHFA6","canonical_record":{"source":{"id":"2112.14790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-12-29T19:03:12Z","cross_cats_sorted":[],"title_canon_sha256":"d817a73b19a2329b954a10f2ee86e03b68fd22449f7d76313205efe1f1ddd476","abstract_canon_sha256":"7615589d361f5f8cd187a0a5bac220ee3ecfb5b1d58d7fb836158d1746ae8b6e"},"schema_version":"1.0"},"canonical_sha256":"b78183941e9a46bd1ff31a1e1249d7ec45f7a432433afdea7525e3d30102ae0c","source":{"kind":"arxiv","id":"2112.14790","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.14790","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"arxiv_version","alias_value":"2112.14790v2","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.14790","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_12","alias_value":"W6AYHFA6TJDL","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_16","alias_value":"W6AYHFA6TJDL2H7T","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_8","alias_value":"W6AYHFA6","created_at":"2026-06-26T01:15:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:W6AYHFA6TJDL2H7TDIPBESOX5R","target":"record","payload":{"canonical_record":{"source":{"id":"2112.14790","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-12-29T19:03:12Z","cross_cats_sorted":[],"title_canon_sha256":"d817a73b19a2329b954a10f2ee86e03b68fd22449f7d76313205efe1f1ddd476","abstract_canon_sha256":"7615589d361f5f8cd187a0a5bac220ee3ecfb5b1d58d7fb836158d1746ae8b6e"},"schema_version":"1.0"},"canonical_sha256":"b78183941e9a46bd1ff31a1e1249d7ec45f7a432433afdea7525e3d30102ae0c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:15:41.196837Z","signature_b64":"LxKmPJq771LluxykcZTjXP8RXqT8xEYeNX8AltzM6Nv02lpZkQ8UJkoWuUpOVEA7eXYjY4yylqtH6mlrI2MwDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b78183941e9a46bd1ff31a1e1249d7ec45f7a432433afdea7525e3d30102ae0c","last_reissued_at":"2026-06-26T01:15:41.196313Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:15:41.196313Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2112.14790","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":"2026-06-26T01:15:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uGicdBoV0YscOMeUUmgwpLCp9KrtyC1bzEvi3Fn+Toe8ifTc44jQsw9IuTbzrdOiUZmT4+wpCHWXiB/D2VqGAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T14:35:41.241155Z"},"content_sha256":"8b7db3fee2254f925923fa9dc6be00b1e3d0f9817065dabe396449e1d3c1a6ad","schema_version":"1.0","event_id":"sha256:8b7db3fee2254f925923fa9dc6be00b1e3d0f9817065dabe396449e1d3c1a6ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:W6AYHFA6TJDL2H7TDIPBESOX5R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Dihedral Linking Invariants","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Elise Catania, Jack Kendrick, Olivia Del Guercio, Patricia Cahn, Sarangoo Chimgee","submitted_at":"2021-12-29T19:03:12Z","abstract_excerpt":"A Fox p-colored knot $K$ in $S^3$ gives rise to a corresponding $p$-fold dihedral branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko. We then extend this alg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14790","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2112.14790/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-26T01:15:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GTdVF1N6xI45vbCplUKKDasFc13OHjs0YRwFzwGgO/wvW97Jt6ISMN0gq4VlcTS/3mPjhwJggLDdw/2lVduGBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T14:35:41.241549Z"},"content_sha256":"06b19931271cf6a0a262e8349b29a62c2f259a35ec051522d1ba57832caabbb0","schema_version":"1.0","event_id":"sha256:06b19931271cf6a0a262e8349b29a62c2f259a35ec051522d1ba57832caabbb0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/bundle.json","state_url":"https://pith.science/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/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":"2026-06-29T14:35:41Z","links":{"resolver":"https://pith.science/pith/W6AYHFA6TJDL2H7TDIPBESOX5R","bundle":"https://pith.science/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/bundle.json","state":"https://pith.science/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W6AYHFA6TJDL2H7TDIPBESOX5R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:W6AYHFA6TJDL2H7TDIPBESOX5R","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":"7615589d361f5f8cd187a0a5bac220ee3ecfb5b1d58d7fb836158d1746ae8b6e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-12-29T19:03:12Z","title_canon_sha256":"d817a73b19a2329b954a10f2ee86e03b68fd22449f7d76313205efe1f1ddd476"},"schema_version":"1.0","source":{"id":"2112.14790","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2112.14790","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"arxiv_version","alias_value":"2112.14790v2","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2112.14790","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_12","alias_value":"W6AYHFA6TJDL","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_16","alias_value":"W6AYHFA6TJDL2H7T","created_at":"2026-06-26T01:15:41Z"},{"alias_kind":"pith_short_8","alias_value":"W6AYHFA6","created_at":"2026-06-26T01:15:41Z"}],"graph_snapshots":[{"event_id":"sha256:06b19931271cf6a0a262e8349b29a62c2f259a35ec051522d1ba57832caabbb0","target":"graph","created_at":"2026-06-26T01:15:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2112.14790/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A Fox p-colored knot $K$ in $S^3$ gives rise to a corresponding $p$-fold dihedral branched cover $M$ of $S^3$ along $K$. The pre-image of the knot $K$ under the covering map is a $\\dfrac{p+1}{2}$-component link $L$ in $M$, and the set of pairwise linking numbers of the components of $L$ is an invariant of $K$. This powerful invariant played a key role in the development of early knot tables, and appears in formulas for many other important knot and manifold invariants. We give an algorithm for computing this invariant for all odd $p$, generalizing an algorithm of Perko. We then extend this alg","authors_text":"Elise Catania, Jack Kendrick, Olivia Del Guercio, Patricia Cahn, Sarangoo Chimgee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-12-29T19:03:12Z","title":"Dihedral Linking Invariants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2112.14790","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":"sha256:8b7db3fee2254f925923fa9dc6be00b1e3d0f9817065dabe396449e1d3c1a6ad","target":"record","created_at":"2026-06-26T01:15:41Z","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":"7615589d361f5f8cd187a0a5bac220ee3ecfb5b1d58d7fb836158d1746ae8b6e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2021-12-29T19:03:12Z","title_canon_sha256":"d817a73b19a2329b954a10f2ee86e03b68fd22449f7d76313205efe1f1ddd476"},"schema_version":"1.0","source":{"id":"2112.14790","kind":"arxiv","version":2}},"canonical_sha256":"b78183941e9a46bd1ff31a1e1249d7ec45f7a432433afdea7525e3d30102ae0c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b78183941e9a46bd1ff31a1e1249d7ec45f7a432433afdea7525e3d30102ae0c","first_computed_at":"2026-06-26T01:15:41.196313Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:15:41.196313Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LxKmPJq771LluxykcZTjXP8RXqT8xEYeNX8AltzM6Nv02lpZkQ8UJkoWuUpOVEA7eXYjY4yylqtH6mlrI2MwDw==","signature_status":"signed_v1","signed_at":"2026-06-26T01:15:41.196837Z","signed_message":"canonical_sha256_bytes"},"source_id":"2112.14790","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8b7db3fee2254f925923fa9dc6be00b1e3d0f9817065dabe396449e1d3c1a6ad","sha256:06b19931271cf6a0a262e8349b29a62c2f259a35ec051522d1ba57832caabbb0"],"state_sha256":"a8b793a5baab2fe757549209bb3a5a5d36fb49bca84ec1d0ad4082e4d14cb502"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"myoWO0RClSoHBXZBygYU7+oTPDPR8Q5H8ts/UaKMjh/ig6Qf6TRHLwXcfeDvyzw582+zo3gY9OUXXMW6pzX6DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T14:35:41.243536Z","bundle_sha256":"2627e7bd89c67a68d032115fbeff4db9a8220b37e3e9a23bc62a9bf700b4a8e7"}}