Self-Dual Maps II: Links and Symmetry

In this paper, we investigate representations of links that are either centrally symmetric in R or antipodally symmetric in S. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [9], we are able to present sufficient combinatorial conditions for a link L to admit such representations. The latter naturally arises sufficient conditions for L to be amphichiral. We also introduce another (closely related) method yielding again to sufficient conditions for L to be amphichiral. We finally prove that a link L, associated to a map G, is amphichiral if the self-dual pairing of G is not one of 6 specific ones among the classification of the 24 self-dual pairing Cor(G)BAut(G).