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For each triple of integers $k,m,n$ such that $m\\equiv n \\pmod{2}$, we explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\\times S^1)\\sqcup S^3 \\rightarrow {\\mathbb R}^6$ such that the following theorem holds.\n  Theorem: Any Brunnian embedding $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is isotopic to $f_{k,m,n}$ for some integers $k,m,n$ such that $m\\equiv n \\pmod{2}$."},"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":"1408.3918","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2014-08-18T08:04:39Z","cross_cats_sorted":[],"title_canon_sha256":"5326d2179ac5a2016344145dfefa9d8abe000566940a0ab7aeca39dc879d6bf7","abstract_canon_sha256":"60c26f769db2a18767a96c9f12815a981dddbe7a0fcdaf8bc61c21f0de3d52af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:08.559285Z","signature_b64":"DYspGTOQa1II0XAlwwslTxd2GJrUhaEV9p8+1BhijwvHn49cXIlUOvDwwnueK/lI/GtOTVWyFFXD01gEJk2WAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5338d0f15cfff86ccd604ee531f28f032d271bb45d7454b81882c4dbf4ee2454","last_reissued_at":"2026-05-18T01:22:08.558741Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:08.558741Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The classification of certain linked $3$-manifolds in $6$-space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Sergey Avvakumov","submitted_at":"2014-08-18T08:04:39Z","abstract_excerpt":"We work entirely in the smooth category. An embedding $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is {\\it Brunnian}, if the restriction of $f$ to each component is isotopic to the standard embedding. For each triple of integers $k,m,n$ such that $m\\equiv n \\pmod{2}$, we explicitly construct a Brunnian embedding $f_{k,m,n}:(S^2\\times S^1)\\sqcup S^3 \\rightarrow {\\mathbb R}^6$ such that the following theorem holds.\n  Theorem: Any Brunnian embedding $f:(S^2\\times S^1)\\sqcup S^3\\rightarrow {\\mathbb R}^6$ is isotopic to $f_{k,m,n}$ for some integers $k,m,n$ such that $m\\equiv n \\pmod{2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.3918","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":""},"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":"1408.3918","created_at":"2026-05-18T01:22:08.558807+00:00"},{"alias_kind":"arxiv_version","alias_value":"1408.3918v2","created_at":"2026-05-18T01:22:08.558807+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.3918","created_at":"2026-05-18T01:22:08.558807+00:00"},{"alias_kind":"pith_short_12","alias_value":"KM4NB4K4774G","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KM4NB4K4774GZTLA","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KM4NB4K4","created_at":"2026-05-18T12:28:35.611951+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/KM4NB4K4774GZTLAJ3STD4UPAM","json":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM.json","graph_json":"https://pith.science/api/pith-number/KM4NB4K4774GZTLAJ3STD4UPAM/graph.json","events_json":"https://pith.science/api/pith-number/KM4NB4K4774GZTLAJ3STD4UPAM/events.json","paper":"https://pith.science/paper/KM4NB4K4"},"agent_actions":{"view_html":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM","download_json":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM.json","view_paper":"https://pith.science/paper/KM4NB4K4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1408.3918&json=true","fetch_graph":"https://pith.science/api/pith-number/KM4NB4K4774GZTLAJ3STD4UPAM/graph.json","fetch_events":"https://pith.science/api/pith-number/KM4NB4K4774GZTLAJ3STD4UPAM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM/action/storage_attestation","attest_author":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM/action/author_attestation","sign_citation":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM/action/citation_signature","submit_replication":"https://pith.science/pith/KM4NB4K4774GZTLAJ3STD4UPAM/action/replication_record"}},"created_at":"2026-05-18T01:22:08.558807+00:00","updated_at":"2026-05-18T01:22:08.558807+00:00"}