Topological Crystals
John C. Baez
TL;DR
This work develops the theory of topological crystals by embedding the maximal abelian cover $\overline{X}$ of a bridgeless graph $X$ into the real first homology space $H_1(X,\mathbb{R})$, yielding a crystal-like pattern with atoms at the cover's vertices and bonds along its edges. It establishes that the embedding exists and is injective precisely when $X$ has no bridges, and that graph symmetries extend to affine isometries of the crystal via the covering-symmetry group ${\rm Cov}(X)$, which fits into a short exact sequence with $H_1(X,\mathbb{Z})$ and ${\rm Aut}(X)$. The packing fraction of the crystal is computed using BHN's integral cycles and cuts, giving a closed form $|V|/|T|$ for finite bridgeless graphs, where $|V|$ is the vertex count and $|T|$ the number of spanning trees. The framework encompasses graphene and diamond as canonical examples and yields a spectrum of highly symmetric, high-dimensional crystals associated with familiar graphs (tetrahedron, cube, Petersen graph, Klein's quartic tilings) along with explicit symmetry-extensions and spanning-tree data. Overall, the paper bridges covering theory, homology, and crystallography to construct and analyze topological crystals from graphs.
Abstract
Sunada's work on crystallography emphasizes the role of the "maximal abelian cover" of a graph $X$. This is a covering space of $X$ for which the group of deck transformations is the first homology group $H_1(X,\mathbb{Z})$. An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie along the edges. We prove that for any connected graph $X$ without bridges, there is a canonical embedding of the maximal abelian cover of $X$ into the vector space $H_1(X,\mathbb{R})$, called a "topological crystal". Crystals of graphene and diamond are examples of this construction. We prove that any symmetry of a graph lifts to a symmetry of its topological crystal. We also compute the density of atoms in a topological crystal. The key technical tools are a way of decomposing the 1-chain coming from a path in $X$ into manageable pieces, and the work of Bacher, de la Harpe and Nagnibeda on integral cycles and integral cuts.