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For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $\\Sigma_g$ of genus $g$ which extends over $(S^3, \\Sigma_g)$, where the maximum is taken over all possible embeddings $\\Sigma_g\\hookrightarrow S^3$. We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$.\n  In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\\ne k^2$ or $4(\\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$."},"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":"1209.1170","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2012-09-06T03:44:09Z","cross_cats_sorted":["math.CO","math.GR"],"title_canon_sha256":"0f0a8ec94fd31f444028389c610032eb5e1b5b0e6e145270e94eaeb27ed5ce27","abstract_canon_sha256":"559967e3f6c02612045e09bb721303099df6b825635913329cf13ddc907af8dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:58.949261Z","signature_b64":"LB9SdR3gkvm+KzjTAr0zkbb2dGDqk9CNt+mJX1nGoGYGiBjOIDPWdTgEyFDtaPOqz9VGiaw335MKMOkCfSysDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e3a467d8a4180b51dfd7421834603fa93ebd4938ec0c75306b351fd90e18f658","last_reissued_at":"2026-05-18T01:12:58.948525Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:58.948525Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Embedding surfaces into $S^3$ with maximum symmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR"],"primary_cat":"math.GT","authors_text":"Bruno Zimmermann, Chao Wang, Shicheng Wang, Yimu Zhang","submitted_at":"2012-09-06T03:44:09Z","abstract_excerpt":"We restrict our discussion to the orientable category. 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We will determine $OE_g$ for each $g$, indeed the action realizing $OE_g$.\n  In particular, with 23 exceptions, $OE_g$ is $4(g+1)$ if $g\\ne k^2$ or $4(\\sqrt{g}+1)^2$ if $g=k^2$, and moreover $OE_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.1170","kind":"arxiv","version":4},"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":"1209.1170","created_at":"2026-05-18T01:12:58.948904+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.1170v4","created_at":"2026-05-18T01:12:58.948904+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.1170","created_at":"2026-05-18T01:12:58.948904+00:00"},{"alias_kind":"pith_short_12","alias_value":"4OSGPWFEDAFV","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"4OSGPWFEDAFVDX6X","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"4OSGPWFE","created_at":"2026-05-18T12:26:53.410803+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/4OSGPWFEDAFVDX6XIIMDIYB7VE","json":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE.json","graph_json":"https://pith.science/api/pith-number/4OSGPWFEDAFVDX6XIIMDIYB7VE/graph.json","events_json":"https://pith.science/api/pith-number/4OSGPWFEDAFVDX6XIIMDIYB7VE/events.json","paper":"https://pith.science/paper/4OSGPWFE"},"agent_actions":{"view_html":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE","download_json":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE.json","view_paper":"https://pith.science/paper/4OSGPWFE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.1170&json=true","fetch_graph":"https://pith.science/api/pith-number/4OSGPWFEDAFVDX6XIIMDIYB7VE/graph.json","fetch_events":"https://pith.science/api/pith-number/4OSGPWFEDAFVDX6XIIMDIYB7VE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE/action/storage_attestation","attest_author":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE/action/author_attestation","sign_citation":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE/action/citation_signature","submit_replication":"https://pith.science/pith/4OSGPWFEDAFVDX6XIIMDIYB7VE/action/replication_record"}},"created_at":"2026-05-18T01:12:58.948904+00:00","updated_at":"2026-05-18T01:12:58.948904+00:00"}