Self-embeddings of doubled affine Steiner triple systems

Abstract

Given a properly face two-coloured triangulation of the graph $K_n$ in a surface, a Steiner triple system can be constructed from each of the colour classes. The two Steiner triple systems obtained in this manner are said to form a biembedding. If the systems are isomorphic to each other it is a self-embedding. In the following, for each $k\geq 2$, we construct a self-embedding of the doubled affine Steiner triple system AG$(k,3)$ in a nonorientable surface. We also make use of a construction due to Grannell, Griggs and Širáň to obtain a biembedding of AG$(k,3)$ in a nonorientable surface that is not a self-embedding for $k>2$.