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This group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting components or reversing orientations; it is a subgroup of $\\Gamma_2$, the group of all such operations.\n  For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of $\\Gamma_2$ up to conjug"},"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":"1201.2722","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-01-13T01:28:32Z","cross_cats_sorted":[],"title_canon_sha256":"74f134e52176ca32330618763c95fef7715ebc70d19a8378e6fe1161e559afea","abstract_canon_sha256":"cc9dd69d7c46fd0e5cdec9aaa0dacb6165cb159ce8bf1117ceb7644177e9baa6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:11.265986Z","signature_b64":"H1FgFROr2bQxrwpyD0DN5xGpnvslz2gmBA1xMaf612zmkyDlrrsCx1od81DtQcfeEmUSMl81aoEM3glwQpdFCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"126e261f13a5d291a0882bdc5879d255e1768c984b5eff6c35fcc3d948a43aa9","last_reissued_at":"2026-05-18T02:25:11.265492Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:11.265492Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 27 possible intrinsic symmetry groups of two-component links","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"James Cornish, Jason Cantarella, Jason Parsley, Matt Mastin","submitted_at":"2012-01-13T01:28:32Z","abstract_excerpt":"We consider the \"intrinsic\" symmetry group of a two-component link $L$, defined to be the image $\\Sigma(L)$ of the natural homomorphism from the standard symmetry group $\\MCG(S^3,L)$ to the product $\\MCG(S^3) \\cross \\MCG(L)$. 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