Embedding surfaces into $S^3$ with maximum symmetry

Abstract

We restrict our discussion to the orientable category. For $g>1$, let ${\mathrm{OE}}_g$ be the maximum order of a finite group $G$ acting on the closed surface ${\Sigma }_g$ of genus $g$ which extends over $(S^3,{\Sigma }_g)$, for all possible embeddings ${\Sigma }_g\hookrightarrow S^3$. We will determine ${\mathrm{OE}}_g$ for each $g$, indeed the action realizing ${\mathrm{OE}}_g$.

In particular, with 23 exceptions, ${\mathrm{OE}}_g$ is $4(g+1)$ if $g\ne k^2$ or $4(\sqrt{g}+1{)}^2$ if $g=k^2$,and moreover ${\mathrm{OE}}_g$ can be realized by unknotted embeddings for all $g$ except for $g=21$ and $481$.