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 $n\geqslant 3$ and $\lambda\geqslant 2$. 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.