Vertex collisions in 3-periodic nets of genus 4

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known. Moreover, examples of unstable nets are still scarce. This article presents a systematic analysis of unstable 3-periodic nets of genus 4, satisfying the restrictions that, in a suitable basis, (i) their labelled quotient graph contains a spanning tree with zero voltage and (ii) voltage coordinates belong to the set {−1, 0, 1}. These nets have been defined by a unique circuit of null voltage in the quotient graph. They have been characterized through a shortest path between colliding vertices. The quotient graph and the nature of the net obtained after identification of colliding vertices, if known, are also provided. The complete list of the respective unstable nets, with a detailed description of the results, can be found in the supporting information.

Non-crystallographic nets: characterization and first steps towards a classification

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroupF(N) of the automorphism group of NC nets (N,T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroupF(N).