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Since the normal closure of u in G(K) coincides with that of the inverse of u, and u and its inverse u correspond to a slope r, a rational number or 1/0, we write << r >> = << u >>. The normal closure << u >> describes elements which are trivialized by r-Dehn filling of E(K). In this article, we prove that << r_1 >> =<< r_2 >> if and only if r_1 = r_2, and for a given finite family of slopes S ="},"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":"1705.04470","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2017-05-12T08:40:18Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"bede3133392bc8f082c5e4fe18b32ec695237cfb95cee57508bd99eae0154787","abstract_canon_sha256":"0573a01f14703ce83863cd940850abb070eaf102beff79709c691f770307f769"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:38.944517Z","signature_b64":"LvYJ2gUnHc2Th38kmgiX/vtuLwz59SKOMA7l3C8K0K9O4gqQQbs6bck6NYKrQ4wuSmqAhorg42/Q7c4N+/mWAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7db6117198d5fa70d421f285a1d0859c355c4f12a6150eb45aa62ba215bf030c","last_reissued_at":"2026-05-17T23:50:38.944075Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:38.944075Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nontrivial elements in a knot group which are trivialized by Dehn fillings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GT","authors_text":"Kimihiko Motegi, Masakazu Teragaito, Tetsuya Ito","submitted_at":"2017-05-12T08:40:18Z","abstract_excerpt":"Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K) coincides with that of the inverse of u, and u and its inverse u correspond to a slope r, a rational number or 1/0, we write << r >> = << u >>. The normal closure << u >> describes elements which are trivialized by r-Dehn filling of E(K). In this article, we prove that << r_1 >> =<< r_2 >> if and only if r_1 = r_2, and for a given finite family of slopes S ="},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04470","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":"1705.04470","created_at":"2026-05-17T23:50:38.944139+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.04470v2","created_at":"2026-05-17T23:50:38.944139+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04470","created_at":"2026-05-17T23:50:38.944139+00:00"},{"alias_kind":"pith_short_12","alias_value":"PW3BC4MY2X5H","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"PW3BC4MY2X5HBVBB","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"PW3BC4MY","created_at":"2026-05-18T12:31:37.085036+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/PW3BC4MY2X5HBVBB6KC2DUEFTQ","json":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ.json","graph_json":"https://pith.science/api/pith-number/PW3BC4MY2X5HBVBB6KC2DUEFTQ/graph.json","events_json":"https://pith.science/api/pith-number/PW3BC4MY2X5HBVBB6KC2DUEFTQ/events.json","paper":"https://pith.science/paper/PW3BC4MY"},"agent_actions":{"view_html":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ","download_json":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ.json","view_paper":"https://pith.science/paper/PW3BC4MY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.04470&json=true","fetch_graph":"https://pith.science/api/pith-number/PW3BC4MY2X5HBVBB6KC2DUEFTQ/graph.json","fetch_events":"https://pith.science/api/pith-number/PW3BC4MY2X5HBVBB6KC2DUEFTQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ/action/storage_attestation","attest_author":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ/action/author_attestation","sign_citation":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ/action/citation_signature","submit_replication":"https://pith.science/pith/PW3BC4MY2X5HBVBB6KC2DUEFTQ/action/replication_record"}},"created_at":"2026-05-17T23:50:38.944139+00:00","updated_at":"2026-05-17T23:50:38.944139+00:00"}