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We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \\cap P_K = \\{e\\}$ for any $g \\in G_K$ with $g \\notin P_K$, unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups"},"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":"1104.3062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-15T14:06:10Z","cross_cats_sorted":[],"title_canon_sha256":"bec153c1b5c8f5e529ae38e865418485402d4b6ffd53f13d1d1b24bccd5896b0","abstract_canon_sha256":"30dc6da2293f6530f6c0cce175ba8e8ba8bfeacc30e736847bfc75f13066011c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:24:06.644443Z","signature_b64":"H6XAOZ0kbhcg5AuMkwzFgdkAJnfuTIT8YgYQWHKdi77j7R4plS68t+ILC1X1nEdwnD0DU0hLNJY6HDUi13J1DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf47e21164d9971ec7894e6c81a784014e42f8dcc0e5ac875772764523d11220","last_reissued_at":"2026-05-18T04:24:06.643975Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:24:06.643975Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On malnormal peripheral subgroups in fundamental groups of 3-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Claude Weber, Pierre de la Harpe","submitted_at":"2011-04-15T14:06:10Z","abstract_excerpt":"Let $K$ be a non-trivial knot in the 3-sphere, $E_K$ its exterior, $G_K = \\pi_1(E_K)$ its group, and $P_K = \\pi_1(\\partial E_K) \\subset G_K$ its peripheral subgroup. We show that $P_K$ is malnormal in $G_K$, namely that $gP_Kg^{-1} \\cap P_K = \\{e\\}$ for any $g \\in G_K$ with $g \\notin P_K$, unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_K$ attached to $T_K$ which are not boundary parallel (Theorem 1 and Corollary 2). 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