Cantankerous maps and rotary embeddings of K_n

In this paper we will define a class of locally non-orientable regular maps called "cantankerous." We will show that cantankerous maps are self-Petrie, we will prove a lower bound on the number of vertices such a map may have, and we will give some data which suggest that the cantankerous maps are a fairly restricted class of regular maps. Our main result here is that any vertex-improper map must either be one of these cantankerous maps or be constructed from a smaller vertex-proper map by the Riemann-surface algorithm. We then apply these results to graph theory. Biggs has shown that if M is an orientable rotary map whose underlying graph is K_n, then n must be a power of a prime. We will show that, if n > 6, K_n has no regular embedding; this shows that the only exception to Biggs' theorem in the non-orientable case is n = 6, and that the rotary embeddings of K_n given by Heffter's construction are chiral.