科研之友
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摘要
In this article, we extend the definition of Christoffel words to directed subgraphs of the hypercubic lattice in an arbitrary dimension that we call Christoffel graphs. Christoffel graphs, when , correspond to the well-known Christoffel words. Due to periodicity, the -dimensional Christoffel graph can be embedded in a -torus (a parallelogram when ). We show that Christoffel graphs have similar properties to those of Christoffel words: symmetry of their central part and conjugation with their reversal. Our main result extends Pirillo's theorem (characterization of Christoffel words which asserts that a word is a Christoffel word if and only if it is conjugate to ) to an arbitrary dimension. In the generalization, the map is seen as a flip operation on graphs embedded in and the conjugation is a translation. We show that a fully periodic subgraph of the hypercubic lattice is a translation of its flip if and only if it is a Christoffel graph.
- 出版日期2015-7
