{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:2MHXV6Q6PBK53S7YCU3SBEG7M7","short_pith_number":"pith:2MHXV6Q6","schema_version":"1.0","canonical_sha256":"d30f7afa1e7855ddcbf815372090df67f3a8ae9ed3db20554bc6308a24b4aa02","source":{"kind":"arxiv","id":"1901.10929","version":1},"attestation_state":"computed","paper":{"title":"Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani, Daniel Cavey","submitted_at":"2019-01-30T16:22:01Z","abstract_excerpt":"By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\\left\\{ \\frac{1}{r}(1,s_{1}), \\frac{1}{r}(1,s_{2}), \\ldots, \\frac{1}{r}(1,s_{k}) \\right\\}$ for $k,r \\in \\mathbb{Z}_{>0}$, and $1 \\leq s_{i} < r$ is coprime to $r$."},"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":"1901.10929","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-01-30T16:22:01Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"e348f1a5a0a68fe3bb6a902348912ddb51b6c4c4940b836927714e6e7708c9e6","abstract_canon_sha256":"336460c90dd02172a3382321c32ccc2faefa3dcc87fe6a4a0a274fc26fb91c68"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:05.708832Z","signature_b64":"LnWLx0BEbf/lB6mv3FRLrwj7Hj4FmVWABvJ+V5HB65vFh/TbvXdtsSQ8v9R4AFunANjev/7QN9GQ8dOv9zW2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d30f7afa1e7855ddcbf815372090df67f3a8ae9ed3db20554bc6308a24b4aa02","last_reissued_at":"2026-05-17T23:55:05.708367Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:05.708367Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Winding Number of $r$-modular sequences and Applications to the Singularity Content of a Fano Polygon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Akihiro Higashitani, Daniel Cavey","submitted_at":"2019-01-30T16:22:01Z","abstract_excerpt":"By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose basket of residual singularities is of the form $\\left\\{ \\frac{1}{r}(1,s_{1}), \\frac{1}{r}(1,s_{2}), \\ldots, \\frac{1}{r}(1,s_{k}) \\right\\}$ for $k,r \\in \\mathbb{Z}_{>0}$, and $1 \\leq s_{i} < r$ is coprime to $r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.10929","kind":"arxiv","version":1},"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":"1901.10929","created_at":"2026-05-17T23:55:05.708447+00:00"},{"alias_kind":"arxiv_version","alias_value":"1901.10929v1","created_at":"2026-05-17T23:55:05.708447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.10929","created_at":"2026-05-17T23:55:05.708447+00:00"},{"alias_kind":"pith_short_12","alias_value":"2MHXV6Q6PBK5","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"2MHXV6Q6PBK53S7Y","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"2MHXV6Q6","created_at":"2026-05-18T12:33:07.085635+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/2MHXV6Q6PBK53S7YCU3SBEG7M7","json":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7.json","graph_json":"https://pith.science/api/pith-number/2MHXV6Q6PBK53S7YCU3SBEG7M7/graph.json","events_json":"https://pith.science/api/pith-number/2MHXV6Q6PBK53S7YCU3SBEG7M7/events.json","paper":"https://pith.science/paper/2MHXV6Q6"},"agent_actions":{"view_html":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7","download_json":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7.json","view_paper":"https://pith.science/paper/2MHXV6Q6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1901.10929&json=true","fetch_graph":"https://pith.science/api/pith-number/2MHXV6Q6PBK53S7YCU3SBEG7M7/graph.json","fetch_events":"https://pith.science/api/pith-number/2MHXV6Q6PBK53S7YCU3SBEG7M7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7/action/storage_attestation","attest_author":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7/action/author_attestation","sign_citation":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7/action/citation_signature","submit_replication":"https://pith.science/pith/2MHXV6Q6PBK53S7YCU3SBEG7M7/action/replication_record"}},"created_at":"2026-05-17T23:55:05.708447+00:00","updated_at":"2026-05-17T23:55:05.708447+00:00"}