Topological Crystals
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2016
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| _version_ | 1866914278221545472 |
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| author | Baez, John C. |
| author_facet | Baez, John C. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1607_07748 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Topological Crystals Baez, John C. Algebraic Topology Combinatorics 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. |
| title | Topological Crystals |
| topic | Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/1607.07748 |