6 - Symmetries of polycycles

Summary

Recall that the automorphism group Aut (P) of an (r, q)-polycycle P is the group of automorphisms of the plane graph preserving the set of interior faces (see Section 4.1). Call a polycycle P isotoxal, isogonal, or isohedral if Aut (P) is transitive on edges, vertices, or interior faces, respectively. In this chapter we first consider the possible automorphism groups of an (r, q)-polycycle, then we list all isogonal or isohedral polycycles for elliptic (r, q) and present a general algorithm for their enumeration. We also present the problem of determining all isogonal and isohedral (r, q)_gen-polycycles.

Automorphism group of (r, q)-polycycles

If an (r, q)-polycycle P is finite, then it has a single boundary and Aut(P) is a dihedral group consisting only of rotations and mirrors around this boundary. So its order divides 2r, 4, or 2q, depending on what Aut(P) fixes: the center of an r-gon, the center of an edge, or a vertex.

None of (3, 3)-, (3, 4)-, (4, 3)-polycycles has, but almost all (r, q)-polycycles for any other (r, q) have, trivial Aut(P).

The number of chiral (i.e. with Aut(P) containing only rotations and translations) proper (5, 3)-, (3, 5)-polycycles is 12, 208 (amongst, respectively, all 39, 263.)

Given an (r, q)-polycycle P, consider the cell-homomorphism f, presented in Section 4.3, from P to {r, q}. It maps the group Aut(P) into Aut({r, q}) = T *(l, m, n). The image φ(Aut(P)) consists of automorphisms of φ(P). If P is a proper polycycle, then Aut(P) coincides with Aut(φ(P)). Otherwise, Aut(P) is an extension of Aut(φ(P)) by the kernel of this homomorphism.