{"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. 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 $481$."},"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"}