Embedding surfaces into $S^3$ with maximum symmetry

TL;DR

This paper determines the maximum order of finite groups acting on embedded surfaces in $S^3$ with maximum symmetry, revealing a pattern related to the genus and identifying exceptions.

Contribution

It explicitly computes the maximum symmetry order for surfaces of genus greater than one and characterizes the embeddings realizing these symmetries.

Findings

Maximum order of symmetry is 4(g+1) for most g

Exceptional cases occur at perfect squares and their roots

Most maximum symmetry actions are realizable by unknotted embeddings

Abstract

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 $OE_g$. 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 $481$.