Self-dual Maps I : antipodality

TL;DR

This paper explores the properties and conditions of self-dual maps that are antipodally symmetric on the sphere, providing combinatorial characterizations and examining their implications for symmetry and geometric problems.

Contribution

It introduces a combinatorial characterization of antipodally self-dual maps using involutive labelings and investigates their relation to antipodal symmetry and convex geometry.

Findings

Characterization of antipodally self-dual maps via involutive labelings

Necessary conditions for strongly involutive maps

Relation between antipodally self-dual and antipodally symmetric maps

Abstract

A self-dual map $G$ is said to be \emph{antipodally self-dual} if the dual map $G^*$ is antipodal embedded in $\mathbb{S}^2$ with respect to $G$. In this paper, we investigate necessary and/or sufficient conditions for a map to be antipodally self-dual. In particular, we present a combinatorial characterization for map $G$ to be antipodally self-dual in terms of certain \emph{involutive labelings}. The latter lead us to obtain necessary conditions for a map to be \emph{strongly involutive} (a notion relevant for its connection with convex geometric problems). We also investigate the relation of antipodally self-dual maps and the notion of \emph{ antipodally symmetric} maps. It turns out that the latter is a very helpful tool to study questions concerning the \emph{symmetry} as well as the \emph{amphicheirality} of \emph{links}.