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Conway Symbols of Frieze Patterns of Dynkin Types

  • Kengo Miyamoto
  • Ryuhei Suzuki
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Abstract

In this paper, we classify the Conway symbols of the Euclidean symmetry groups of frieze patterns of Dynkin types An, Bn, Cn, Dn, F4 and G2. First, we show that the Conway symbol of a Conway--Coxeter frieze is determined by the bilateral and the central symmetry of the underlying triangulation. Secondly, we show that a frieze pattern of non-simply-laced type has the symbol *∞∞ or ∞∞ according as its first interior row is palindromic or not, with the single exception of type C3, which realises four symbols. Thirdly, we show that the same dichotomy holds for type Dn with n ≥ 5, the symbol being read off the tagged triangulation of the once-punctured polygon attached to the frieze together with the value on its puncture edges, and that type D4 realises five symbols.

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Cite

BibTeXRIS
BibTeX
@misc{miyamoto2026conway,
  title = {{Conway Symbols of Frieze Patterns of Dynkin Types}},
  author = {Kengo Miyamoto and Ryuhei Suzuki},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.14948},
  eprint = {2609.14948},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.14948},
}
RIS
TY  - UNPB
TI  - Conway Symbols of Frieze Patterns of Dynkin Types
AU  - Kengo Miyamoto
AU  - Ryuhei Suzuki
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.14948
UR  - https://arxiv.org/abs/2609.14948
AB  - In this paper, we classify the Conway symbols of the Euclidean symmetry groups of frieze patterns of Dynkin types A_n, B_n, C_n, D_n, F_4 and G_2. First, we show that the Conway symbol of a Conway--Coxeter frieze is determined by the bilateral and the central symmetry of the underlying triangulation. Secondly, we show that a frieze pattern of non-simply-laced type has the symbol *∞∞ or ∞∞ according as its first interior row is palindromic or not, with the single exception of type C_3, which realises four symbols. Thirdly, we show that the same dichotomy holds for type D_n with n ≥ 5, the symbol being read off the tagged triangulation of the once-punctured polygon attached to the frieze together with the value on its puncture edges, and that type D_4 realises five symbols.
ER  -