SELF-DUAL MAPS I: ANTIPODALITY

Abstract

A self-dual map G is said to be antipodally self-dual if the dual map G\ast is antipodal embedded in \BbbS 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 involutive labelings. The latter lead us to obtain necessary conditions for a map to be 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 antipodally symmetric maps. It turns out that the latter is a very helpful tool to study questions concerning the symmetry as well as the amphicheirality of links.