{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:CQEHISAJL23E46KWXUERNAKELI","short_pith_number":"pith:CQEHISAJ","canonical_record":{"source":{"id":"1504.03368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-13T21:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"e3ed78358e261b4f7f4dfb8ff635b2209a486d57f0f4aac498c5e9ac1369556a","abstract_canon_sha256":"3f91b540f7793d586a3ac6a7f85f51b7276f4ac16a5106d54f649f1505007365"},"schema_version":"1.0"},"canonical_sha256":"14087448095eb64e7956bd091681445a37d2e36142d0c0917d230d1c60a4d902","source":{"kind":"arxiv","id":"1504.03368","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03368","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03368v1","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03368","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"pith_short_12","alias_value":"CQEHISAJL23E","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CQEHISAJL23E46KW","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CQEHISAJ","created_at":"2026-05-18T12:29:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:CQEHISAJL23E46KWXUERNAKELI","target":"record","payload":{"canonical_record":{"source":{"id":"1504.03368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-13T21:39:56Z","cross_cats_sorted":[],"title_canon_sha256":"e3ed78358e261b4f7f4dfb8ff635b2209a486d57f0f4aac498c5e9ac1369556a","abstract_canon_sha256":"3f91b540f7793d586a3ac6a7f85f51b7276f4ac16a5106d54f649f1505007365"},"schema_version":"1.0"},"canonical_sha256":"14087448095eb64e7956bd091681445a37d2e36142d0c0917d230d1c60a4d902","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:05.187628Z","signature_b64":"yrjpClH3BV/gA1T2a30naEBfOsAUN8ZL1N7CWZWf98iqHoC+bA7ahiZcfXTcj5QYFutQ6vMIPYat8A5Kb1l1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14087448095eb64e7956bd091681445a37d2e36142d0c0917d230d1c60a4d902","last_reissued_at":"2026-05-18T01:02:05.186957Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:05.186957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.03368","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":"2026-05-18T01:02:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yP1sGIiaWnfIe9vRiPLKS2y1Ge8qDUznXrG1xunxEyfUF4YQag+71gOzjJlH5f+RS93dnSUE8x8U95+3pZY6CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:36:19.140516Z"},"content_sha256":"73da3a95817a0cc5bbae2f3c93fee731276c1512078256e66edfed129b27a51d","schema_version":"1.0","event_id":"sha256:73da3a95817a0cc5bbae2f3c93fee731276c1512078256e66edfed129b27a51d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:CQEHISAJL23E46KWXUERNAKELI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Doubly slice knots with low crossing number","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Charles Livingston, Jeffrey Meier","submitted_at":"2015-04-13T21:39:56Z","abstract_excerpt":"A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the dou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03368","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":"2026-05-18T01:02:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qZw8y/EbJueq22wYFDThaU3Z9Y0tqJQ4RSb7bUx3kPhD8BxwT36WMshEA4fCejlFxPZ4mxwtb541cSUkyA6zCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:36:19.140867Z"},"content_sha256":"8a3a5bf4ea4929122f08fa6974d96bcca0bdf6f07f40533efa1b1dee919df41f","schema_version":"1.0","event_id":"sha256:8a3a5bf4ea4929122f08fa6974d96bcca0bdf6f07f40533efa1b1dee919df41f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CQEHISAJL23E46KWXUERNAKELI/bundle.json","state_url":"https://pith.science/pith/CQEHISAJL23E46KWXUERNAKELI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CQEHISAJL23E46KWXUERNAKELI/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":"2026-06-02T16:36:19Z","links":{"resolver":"https://pith.science/pith/CQEHISAJL23E46KWXUERNAKELI","bundle":"https://pith.science/pith/CQEHISAJL23E46KWXUERNAKELI/bundle.json","state":"https://pith.science/pith/CQEHISAJL23E46KWXUERNAKELI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CQEHISAJL23E46KWXUERNAKELI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CQEHISAJL23E46KWXUERNAKELI","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":"3f91b540f7793d586a3ac6a7f85f51b7276f4ac16a5106d54f649f1505007365","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-13T21:39:56Z","title_canon_sha256":"e3ed78358e261b4f7f4dfb8ff635b2209a486d57f0f4aac498c5e9ac1369556a"},"schema_version":"1.0","source":{"id":"1504.03368","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.03368","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"arxiv_version","alias_value":"1504.03368v1","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.03368","created_at":"2026-05-18T01:02:05Z"},{"alias_kind":"pith_short_12","alias_value":"CQEHISAJL23E","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_16","alias_value":"CQEHISAJL23E46KW","created_at":"2026-05-18T12:29:17Z"},{"alias_kind":"pith_short_8","alias_value":"CQEHISAJ","created_at":"2026-05-18T12:29:17Z"}],"graph_snapshots":[{"event_id":"sha256:8a3a5bf4ea4929122f08fa6974d96bcca0bdf6f07f40533efa1b1dee919df41f","target":"graph","created_at":"2026-05-18T01:02:05Z","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":"A knot in the three-sphere is doubly slice if it is the cross-section of an unknotted two-sphere in the four-sphere. For low-crossing knots, the most complete work to date gives a classification of doubly slice knots through 9 crossings. We extend that work through 12 crossings, resolving all but four cases among the 2,977 prime knots in that range. The techniques involved in this analysis include considerations of the Alexander module and signature functions as well as calculations of the twisted Alexander polynomials for higher-order branched covers. We give explicit illustrations of the dou","authors_text":"Charles Livingston, Jeffrey Meier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-13T21:39:56Z","title":"Doubly slice knots with low crossing number"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03368","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":"sha256:73da3a95817a0cc5bbae2f3c93fee731276c1512078256e66edfed129b27a51d","target":"record","created_at":"2026-05-18T01:02:05Z","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":"3f91b540f7793d586a3ac6a7f85f51b7276f4ac16a5106d54f649f1505007365","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-13T21:39:56Z","title_canon_sha256":"e3ed78358e261b4f7f4dfb8ff635b2209a486d57f0f4aac498c5e9ac1369556a"},"schema_version":"1.0","source":{"id":"1504.03368","kind":"arxiv","version":1}},"canonical_sha256":"14087448095eb64e7956bd091681445a37d2e36142d0c0917d230d1c60a4d902","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14087448095eb64e7956bd091681445a37d2e36142d0c0917d230d1c60a4d902","first_computed_at":"2026-05-18T01:02:05.186957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:02:05.186957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yrjpClH3BV/gA1T2a30naEBfOsAUN8ZL1N7CWZWf98iqHoC+bA7ahiZcfXTcj5QYFutQ6vMIPYat8A5Kb1l1Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:02:05.187628Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.03368","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73da3a95817a0cc5bbae2f3c93fee731276c1512078256e66edfed129b27a51d","sha256:8a3a5bf4ea4929122f08fa6974d96bcca0bdf6f07f40533efa1b1dee919df41f"],"state_sha256":"833e5646c0c06caf1704a7ebf2ccc0a0bccd07fe7c896c62c8e33f2a86a59939"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HP5CsIBiSQLJM9SkucYCBdajJmgTg9WFVJUqZxDviHRZY6R6R38/Ig72sG7DVNpn/3Oa6zUY+aslBHdetVKZAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T16:36:19.142847Z","bundle_sha256":"a3509440a4d4928a02ae7a82ffe05c3846c7850bbe94c1e71ff50584dab96937"}}