Modelling biplanes on surfaces
A biplane is a symmetric design on \(v = k(k-1)/2 + 1\) points and the same number of lines such that (1) each line contains exactly \(k\) points, (2) each point is on exactly \(k\) lines, and (3) any pair of points lie on exactly 2 lines. Non-trivial biplanes are known to exist only for \(k = 3,4,5,6,9,11\), and \(13\). Biplanes can be generated by 2-fold difference sets. In this context, the biplanes can be modeled by embeddings of Cayley graphs on a possibly nonorientable surface or pseudosurface. In this paper the author gives such models for the seven known biplanes generated by difference sets. He asks for suitable topological models for the remaining 10 known biplanes.
scientific article; zbMATH DE number 2105724