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Then $K$ is equivalent to $J$ if and only if a Seifert matrix associated with a simple Seifert hypersurface for $K$ is $(-1)^p$-$S$-equivalent to that for $J$.\n  We also discuss the $2p+1=3$ case. This result implies one of our main results: Let $\\mu$ be a natural number. A 1-link $A$ is pass-move equivalent to a 1-link $B$ if and only if the knot product of $A$ and $\\mu$ "},"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":"1504.01229","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2015-04-06T07:51:19Z","cross_cats_sorted":[],"title_canon_sha256":"0fd59306782a630af48a2b24fa5244797cb92bcb8baa7c182275acee64123139","abstract_canon_sha256":"9e18186bb2071cec132d11e087d5e43c6f050fbd65de8b13914024d46839556d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:23.315444Z","signature_b64":"iM+UMI2RZP9dIvXcelCWzozw1vEtk2RObAAPzswdVfqS/6e0E7YaRvGlGN/UPG7UqsYHVwdmNrKUnl7cndkeAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd2f797fc53170caa3387be557bbb25acf903855194f815b3d44b82b5ca10825","last_reissued_at":"2026-05-18T00:14:23.314756Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:23.314756Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Brieskorn submanifolds, Local moves on knots, and knot products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Eiji Ogasa, Louis H. Kauffman","submitted_at":"2015-04-06T07:51:19Z","abstract_excerpt":"We prove the following: Let $2p + 1$ be no less than 5 and $p$ be a natural number. Let $K$ and $J$ be closed, oriented, $(2p+1)$-dimensional connected, $(p-1)$-connected, simple submanifolds of the standard $(2p+3)$-sphere. Then $K$ is equivalent to $J$ if and only if a Seifert matrix associated with a simple Seifert hypersurface for $K$ is $(-1)^p$-$S$-equivalent to that for $J$.\n  We also discuss the $2p+1=3$ case. This result implies one of our main results: Let $\\mu$ be a natural number. 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