{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:6KUQMRJOD3NEM5FC56WM7N4I3G","short_pith_number":"pith:6KUQMRJO","schema_version":"1.0","canonical_sha256":"f2a906452e1eda4674a2efaccfb788d9a879de29d3bf9b4b806197e316bec5be","source":{"kind":"arxiv","id":"2506.21130","version":2},"attestation_state":"computed","paper":{"title":"An Invariant for Triple-Point-Free Immersed Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jona Seidel","submitted_at":"2025-06-26T10:12:30Z","abstract_excerpt":"We define an invariant of triple-point-free immersions of $2$-spheres into Euclidean $3$-space, taking values in $l^1(\\mathbb{Z})$. It remains unchanged under regular homotopies through such immersions. An explicit description of its image shows that the space of triple-point-free immersed spheres has infinitely many regular homotopy classes. Consequently, many pairs of immersed spheres can only be connected by regular homotopies that pass through triple points. We represent the double points of a triple-point-free immersed sphere using a directed tree, equipped with a pair relation on the edg"},"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":"2506.21130","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2025-06-26T10:12:30Z","cross_cats_sorted":[],"title_canon_sha256":"f1ba3580da90bb303e93df365bc92dbb09322e3c289cf3bba8e194a074ba85b6","abstract_canon_sha256":"8428dd00dd52065935b16cc6e169f50fcad485f287c69082db155df831dafe81"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:30:02.738767Z","signature_b64":"waBLgVuNLMNp0jllm15fM8RIjRP4VqZL69nr3yEbBV88HX225T/N1ZRCf2/SXQAPqVRokwS8JtUJD06tH2+KBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2a906452e1eda4674a2efaccfb788d9a879de29d3bf9b4b806197e316bec5be","last_reissued_at":"2026-07-05T11:30:02.738270Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:30:02.738270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Invariant for Triple-Point-Free Immersed Spheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jona Seidel","submitted_at":"2025-06-26T10:12:30Z","abstract_excerpt":"We define an invariant of triple-point-free immersions of $2$-spheres into Euclidean $3$-space, taking values in $l^1(\\mathbb{Z})$. It remains unchanged under regular homotopies through such immersions. An explicit description of its image shows that the space of triple-point-free immersed spheres has infinitely many regular homotopy classes. Consequently, many pairs of immersed spheres can only be connected by regular homotopies that pass through triple points. We represent the double points of a triple-point-free immersed sphere using a directed tree, equipped with a pair relation on the edg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.21130","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.21130/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.21130","created_at":"2026-07-05T11:30:02.738336+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.21130v2","created_at":"2026-07-05T11:30:02.738336+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.21130","created_at":"2026-07-05T11:30:02.738336+00:00"},{"alias_kind":"pith_short_12","alias_value":"6KUQMRJOD3NE","created_at":"2026-07-05T11:30:02.738336+00:00"},{"alias_kind":"pith_short_16","alias_value":"6KUQMRJOD3NEM5FC","created_at":"2026-07-05T11:30:02.738336+00:00"},{"alias_kind":"pith_short_8","alias_value":"6KUQMRJO","created_at":"2026-07-05T11:30:02.738336+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.23359","citing_title":"The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow","ref_index":62,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G","json":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G.json","graph_json":"https://pith.science/api/pith-number/6KUQMRJOD3NEM5FC56WM7N4I3G/graph.json","events_json":"https://pith.science/api/pith-number/6KUQMRJOD3NEM5FC56WM7N4I3G/events.json","paper":"https://pith.science/paper/6KUQMRJO"},"agent_actions":{"view_html":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G","download_json":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G.json","view_paper":"https://pith.science/paper/6KUQMRJO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.21130&json=true","fetch_graph":"https://pith.science/api/pith-number/6KUQMRJOD3NEM5FC56WM7N4I3G/graph.json","fetch_events":"https://pith.science/api/pith-number/6KUQMRJOD3NEM5FC56WM7N4I3G/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G/action/storage_attestation","attest_author":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G/action/author_attestation","sign_citation":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G/action/citation_signature","submit_replication":"https://pith.science/pith/6KUQMRJOD3NEM5FC56WM7N4I3G/action/replication_record"}},"created_at":"2026-07-05T11:30:02.738336+00:00","updated_at":"2026-07-05T11:30:02.738336+00:00"}