{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:O4OHPDDWYR3QIK74MIA7TYCCHO","short_pith_number":"pith:O4OHPDDW","schema_version":"1.0","canonical_sha256":"771c778c76c477042bfc6201f9e0423b842b16c186f46e57ad33220c914ed223","source":{"kind":"arxiv","id":"1101.5147","version":4},"attestation_state":"computed","paper":{"title":"On the commutator of unit quaternions and the numbers 12 and 24","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.DG"],"primary_cat":"math.GT","authors_text":"Thomas Puettmann","submitted_at":"2011-01-26T20:29:19Z","abstract_excerpt":"The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \\pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction o"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1101.5147","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-01-26T20:29:19Z","cross_cats_sorted":["math.AT","math.DG"],"title_canon_sha256":"7992a49dd8968335b55bc945b36b9ae7d4c2be588e4cf977c8789e5faf6a883b","abstract_canon_sha256":"1aea8acaf4df2fc53ac115074f26aab37352588364ba25270f9b37fdeef177ae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:54:49.513800Z","signature_b64":"STNBNgcg+YJejDbgtD+BG5g5hJOXRgpxp+CRPPPPtLMAVDZ3tOzdIYChOhvNpw7qpBKV2CvUQKPP5mYMGodtCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"771c778c76c477042bfc6201f9e0423b842b16c186f46e57ad33220c914ed223","last_reissued_at":"2026-05-18T03:54:49.513115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:54:49.513115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the commutator of unit quaternions and the numbers 12 and 24","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT","math.DG"],"primary_cat":"math.GT","authors_text":"Thomas Puettmann","submitted_at":"2011-01-26T20:29:19Z","abstract_excerpt":"The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power of the commutator. Subsequently, we construct free S^3-actions on S^7 x S^3 whose quotients are exotic 7-sphere and give a geometric explanation for the order of the stable homotopy groups \\pi_{n+3} (S^n). Intermediate results of perhaps independent interest are a construction o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5147","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1101.5147","created_at":"2026-05-18T03:54:49.513217+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.5147v4","created_at":"2026-05-18T03:54:49.513217+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.5147","created_at":"2026-05-18T03:54:49.513217+00:00"},{"alias_kind":"pith_short_12","alias_value":"O4OHPDDWYR3Q","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"O4OHPDDWYR3QIK74","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"O4OHPDDW","created_at":"2026-05-18T12:26:37.096874+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO","json":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO.json","graph_json":"https://pith.science/api/pith-number/O4OHPDDWYR3QIK74MIA7TYCCHO/graph.json","events_json":"https://pith.science/api/pith-number/O4OHPDDWYR3QIK74MIA7TYCCHO/events.json","paper":"https://pith.science/paper/O4OHPDDW"},"agent_actions":{"view_html":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO","download_json":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO.json","view_paper":"https://pith.science/paper/O4OHPDDW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.5147&json=true","fetch_graph":"https://pith.science/api/pith-number/O4OHPDDWYR3QIK74MIA7TYCCHO/graph.json","fetch_events":"https://pith.science/api/pith-number/O4OHPDDWYR3QIK74MIA7TYCCHO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO/action/storage_attestation","attest_author":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO/action/author_attestation","sign_citation":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO/action/citation_signature","submit_replication":"https://pith.science/pith/O4OHPDDWYR3QIK74MIA7TYCCHO/action/replication_record"}},"created_at":"2026-05-18T03:54:49.513217+00:00","updated_at":"2026-05-18T03:54:49.513217+00:00"}