Cut-and-project graphs and other complexes

The cut-and-project method may be applied to graphs and complexes, although there are technical difficulties, most notably collisions, i.e. when the images (under the projection) of two disjoint edges or cells intersect. Given a particular periodic structure, a particular projection space, an appropriate window in the orthogonal complement of that space, the induced substructure within the cartesian product of the window and the projection space is projected to the projection space to produce a model structure. We may use an index space of vectors from the orthogonal complement to move the window around and obtain a model system of model structures. We adapt the notion of general position to higher dimensions: the projection space is in doubly general position with respect to a graph or complex when the projection of that structure maps vertices of that structure injectively into the projection space and the edges or polytopes of that structure injectively so that their dimension is not reduced and disjoint edges and polytopes remain disjoint. We find that if the initial structure was a periodic graph, if the projection space and its complement are in general position with respect to the vertices and edges, and the window is also in general position, then the model system has uncountably many isomorphism classes - with distinct coordination sequences. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

The cut-and-project method can be applied to graphs and complexes, but technical difficulties arise, particularly collisions. By projecting the substructure to the projection space, a model structure can be obtained. If all structures are in general position, the model system will have uncountably many isomorphism classes.