Dihedral Groups of Transnormal Embeddings of S1 in Euclidean Spaces

Abstract

Through this work, dihedral groups are exploited to study the transnormality of S1 in Euclidean spaces. We show that generating polytopes of such embeddings follow the transitive action deduced by the fundamental group of the normal plane at the image of S1. We take advantage of the orientability of S1 and the spliting of the generating frame into two sets of the same cardinality, each lies in one of
two parallel planes. We show that if k is odd, k ≥ 3, then 2k-transnormal embeddings of S1 in Rn exist with regular right prisms generating polytopes.