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(R,k)-spheres admitting \\kappa_i<0 are much harder to study. We consider the symmetries and construction for three new instances of such spheres: ({a,b},k)-spheres with p_b\\le 3 (they are listed), icosahedrites (i.e., ({3,4},5)$-spheres) and, for any c\\in N, fullerene c-disks, i.e., ({5,6,c},3)-spheres with p_c=1."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1112.3320","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-14T20:15:03Z","cross_cats_sorted":[],"title_canon_sha256":"81c1052472686bc3ba25d3e37a20af5973757e4d8a7589758cea4cc031560c6d","abstract_canon_sha256":"1df8c18e80abb0b76973692dede5f56ba88b2ae2eb4fd20d0466c902530da505"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:06:19.509753Z","signature_b64":"Yxg+R6o5+w+3CeDG1VmxtYv5snFbbyaJohCPL2bCzhN1iqbKTE2kHt2+npIn5gQCHfM3dPk/OtUAzBkHzcKJDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99bbff26010bef2f98117963b62c525cdbf6735e4b558184a1a5edcd0f930f44","last_reissued_at":"2026-05-18T04:06:19.509101Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:06:19.509101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fullerene-like spheres with faces of negative curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mathieu Dutour Sikiric, Michel Deza, Mikhail Shtogrin","submitted_at":"2011-12-14T20:15:03Z","abstract_excerpt":"Given R\\subset N, an (R,k)$-sphere is a k-regular map on the sphere whose faces have gonalities i\\in R. The most interesting/useful are (geometric) fullerenes, i.e., (\\{5,6\\},3)$-spheres.\n  Call \\kappa_i=1 + \\frac{i}{k} - \\frac{i}{2} the curvature of i-gonal faces. (R,k)-spheres admitting \\kappa_i<0 are much harder to study. We consider the symmetries and construction for three new instances of such spheres: ({a,b},k)-spheres with p_b\\le 3 (they are listed), icosahedrites (i.e., ({3,4},5)$-spheres) and, for any c\\in N, fullerene c-disks, i.e., ({5,6,c},3)-spheres with p_c=1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3320","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":"1112.3320","created_at":"2026-05-18T04:06:19.509203+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.3320v1","created_at":"2026-05-18T04:06:19.509203+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3320","created_at":"2026-05-18T04:06:19.509203+00:00"},{"alias_kind":"pith_short_12","alias_value":"TG576JQBBPXS","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"TG576JQBBPXS7GAR","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"TG576JQB","created_at":"2026-05-18T12:26:42.757692+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/TG576JQBBPXS7GARPFR3MLCSLT","json":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT.json","graph_json":"https://pith.science/api/pith-number/TG576JQBBPXS7GARPFR3MLCSLT/graph.json","events_json":"https://pith.science/api/pith-number/TG576JQBBPXS7GARPFR3MLCSLT/events.json","paper":"https://pith.science/paper/TG576JQB"},"agent_actions":{"view_html":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT","download_json":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT.json","view_paper":"https://pith.science/paper/TG576JQB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.3320&json=true","fetch_graph":"https://pith.science/api/pith-number/TG576JQBBPXS7GARPFR3MLCSLT/graph.json","fetch_events":"https://pith.science/api/pith-number/TG576JQBBPXS7GARPFR3MLCSLT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT/action/storage_attestation","attest_author":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT/action/author_attestation","sign_citation":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT/action/citation_signature","submit_replication":"https://pith.science/pith/TG576JQBBPXS7GARPFR3MLCSLT/action/replication_record"}},"created_at":"2026-05-18T04:06:19.509203+00:00","updated_at":"2026-05-18T04:06:19.509203+00:00"}