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We use this information to demonstrate that thin position is a minimal bridge position of J if and only if the same is true for K, and more generally we show that any thin position of K is an \"obvious\" cabling of a thin position of J. We conclude by proving that J is meridionally small (mp-small) if and only if K is meridionally small (mp-small), and if J is mp-small and every non-minimal bridge position of J is"},"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":"1010.3220","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2010-10-15T16:51:31Z","cross_cats_sorted":[],"title_canon_sha256":"83a3fa89d67da39dace15970bf7b8b424622ce08d17129d4c583bf7a21283916","abstract_canon_sha256":"2038c3a7656afcb9d6bf15d86fb934f0a7311baef1065ffa17c64dd72eae0198"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:17.369661Z","signature_b64":"MYAsE7b4y7xYvbEnvUuW1ZOQhgWXYp5rBqyBWltTWWt69xQ/zPwT1UKNfdw/VB3rdva6u0+m1wrELzvxo55TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0152366822d09032ad36899f55b517b9f5a8147ca5fd0c1c1af05f050054e165","last_reissued_at":"2026-05-18T04:39:17.369071Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:17.369071Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Properties of knots preserved by cabling","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Alexander Zupan","submitted_at":"2010-10-15T16:51:31Z","abstract_excerpt":"We examine geometric properties of a knot J that are unchanged by taking a (p,q)-cable K of J. 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