Bordered surfaces in the 3-sphere with maximum symmetry

TL;DR

This paper determines the maximum symmetry group orders for bordered surfaces embedded in the 3-sphere, classifies their topological types, and extends previous work from closed to bordered surfaces.

Contribution

It extends the classification of maximum symmetry actions from closed surfaces to bordered surfaces in the 3-sphere, including non-orientable cases and algebraic genus considerations.

Findings

Maximum order of symmetry group is 12(α - 1) for specific algebraic genera.

Classified topological types and embeddings of bordered surfaces with maximum symmetry.

Identified specific algebraic genera where maximum symmetry is achieved.

Abstract

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable surfaces, determined for each genus $g>1$ the maximum order of such a $G$ for all embeddings of a surface of genus $g$, and classified the corresponding embeddings. In the present paper we obtain analogous results for the case of bordered surfaces $\Sigma$ (i.e. with non-empty boundary, orientable or not). Now the genus $g$ gets replaced by the algebraic genus $\alpha$ of $\Sigma$ (the rank of its free fundamental group); for each $\alpha > 1$ we determine the maximum order $m_\alpha$ of an action of $G$, classify the topological types of the corresponding surfaces (topological genus, number of boundary components, orientability) and their embeddings into $S^3$. For example, the maximal possibility $12(\alpha - 1)$ is obtained for the finitely many values $\alpha = 2, 3, 4, 5, 9, 11, 25, 97, 121$ and $241$.