{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:YPBZ2MTCVMBL2Y5NS3YDZJGEBM","short_pith_number":"pith:YPBZ2MTC","schema_version":"1.0","canonical_sha256":"c3c39d3262ab02bd63ad96f03ca4c40b076640337a7ab82c4301a2ae7d9cb81f","source":{"kind":"arxiv","id":"1602.09093","version":3},"attestation_state":"computed","paper":{"title":"Orbits of non-simple closed curves on a surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jenya Sapir","submitted_at":"2016-02-29T18:49:23Z","abstract_excerpt":"The mapping class group of a surface $\\S$ acts on the set of closed geodesics on $\\S$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \\geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits that contain geodesics of length at most $L$, for each $L >0$. This result complements a recent result of Mirzakhani, which gives the asymptotic growth of the number of closed geodesics of length at most $L$ in a single mapping class group orbit. Furthermore, we develop a new, c"},"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":"1602.09093","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2016-02-29T18:49:23Z","cross_cats_sorted":[],"title_canon_sha256":"9afffe9f84ca80803bcb5de60b48f4a1801f1d58c8b04987ec11ea0e9ad1f619","abstract_canon_sha256":"5b52e32fe3f214e914630b549086792fbb3f9ce3ce967da74111d24510da9d65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:55.408966Z","signature_b64":"OHOTFalqisD8Eae/Fbjyt+BnNG0xgGttXbTMJSqg1jj5seHEFLfAXHjdsLmSKNh5VEzx8mcblRfh0GIOkY7QCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3c39d3262ab02bd63ad96f03ca4c40b076640337a7ab82c4301a2ae7d9cb81f","last_reissued_at":"2026-05-18T01:10:55.408426Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:55.408426Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbits of non-simple closed curves on a surface","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"Jenya Sapir","submitted_at":"2016-02-29T18:49:23Z","abstract_excerpt":"The mapping class group of a surface $\\S$ acts on the set of closed geodesics on $\\S$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \\geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits that contain geodesics of length at most $L$, for each $L >0$. This result complements a recent result of Mirzakhani, which gives the asymptotic growth of the number of closed geodesics of length at most $L$ in a single mapping class group orbit. Furthermore, we develop a new, c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.09093","kind":"arxiv","version":3},"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":"1602.09093","created_at":"2026-05-18T01:10:55.408541+00:00"},{"alias_kind":"arxiv_version","alias_value":"1602.09093v3","created_at":"2026-05-18T01:10:55.408541+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.09093","created_at":"2026-05-18T01:10:55.408541+00:00"},{"alias_kind":"pith_short_12","alias_value":"YPBZ2MTCVMBL","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_16","alias_value":"YPBZ2MTCVMBL2Y5N","created_at":"2026-05-18T12:30:53.716459+00:00"},{"alias_kind":"pith_short_8","alias_value":"YPBZ2MTC","created_at":"2026-05-18T12:30:53.716459+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/YPBZ2MTCVMBL2Y5NS3YDZJGEBM","json":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM.json","graph_json":"https://pith.science/api/pith-number/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/graph.json","events_json":"https://pith.science/api/pith-number/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/events.json","paper":"https://pith.science/paper/YPBZ2MTC"},"agent_actions":{"view_html":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM","download_json":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM.json","view_paper":"https://pith.science/paper/YPBZ2MTC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1602.09093&json=true","fetch_graph":"https://pith.science/api/pith-number/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/graph.json","fetch_events":"https://pith.science/api/pith-number/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/action/storage_attestation","attest_author":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/action/author_attestation","sign_citation":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/action/citation_signature","submit_replication":"https://pith.science/pith/YPBZ2MTCVMBL2Y5NS3YDZJGEBM/action/replication_record"}},"created_at":"2026-05-18T01:10:55.408541+00:00","updated_at":"2026-05-18T01:10:55.408541+00:00"}