7 - Elementary polycycles

Summary

We have seen in Section 4.5 a full classification of (3, 3)_gen-, (3, 4)_gen-, and (4, 3)_gen polycycles. We have also seen that, for all other (r, q), there is a continuum of (r, q)-polycycles. The purpose of this chapter is to introduce a decomposition of polycycles into elementary components in an analogous way to decompose the molecules into atoms. This method will prove to be very effective but only in the elliptic case, since, for all other cases, we will show that there is a continuum of such elementary components (see Theorem 7.2.1). The first occurrence of the method is in, followed by and.

Decomposition of polycycles

Given an integer q > 3 and a set R ⊂ − {1} (so, 2-gons will be permitted in this chapter), a (R, q)_gen-polycycle is a non-empty 2-connected map on a surface S with faces partitioned in two non-empty sets F₁ and F₂, so it holds that:

1. (i) all elements of F₁ (called proper faces) are combinatorial i-gons with i ∈ R;

2. (ii) all elements of F₂ (called holes, the exterior face(s) are amongst them) are pairwisely disjoint, i.e. have no common vertices;

3. (iii) all vertices have degree within {2, …, q} and all interior (i.e. not on the boundary of a hole) vertices are q-valent.

The map can be finite or infinite and some holes can be i-gons with i ∈ R. If R = {r}, then the above definition corresponds to (r, q)_gen-polycycles.