A classification of rotary embeddings of multicycles

Abstract

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle ${\mathrm{C}}_n^{(\lambda )}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n⩾3$ and $\lambda ⩾2$. When $n$ is odd, there is a unique isomorphism class; when $n$ is even, the embeddings form a family ${\mathcal{M}}_n^{(\lambda )}(i,j)$ parameterized by integer pairs $(i,j)$ satisfying explicit congruence conditions.

arXiv commentWe have decided to withdraw this preprint because we discovered that the main results have already been covered in a previous work by K. Hu et al. Although our study was conducted independently and the methods differ to some extent, the core findings overlap significantly, and we do not believe the methodological novelty is sufficient to warrant a separate publication