A Conway-Coxeter theorem for decorated frieze patterns
Abstract
Let m≥0 and put n=m+3. We introduce a normalized positive Laurent class of decorated frieze patterns and prove a decorated analogue of the Conway-Coxeter classification theorem. Namely, such friezes of width m are in canonical bijection with weighted triangulations of a convex n-gon, whose boundary edges are labeled by independent variables y1,…,yn and whose diagonals are labeled by x1,…,xm. While a weighted Conway-Coxeter propagation algorithm constructs the frieze from a weighted triangulation, a main new ingredient is an explicit nonrecursive Laurent formula expressing each quiddity entry directly from the weighted triangles incident to the corresponding vertex. Conversely, specialization to $1$, Laurent positivity, and a decorated cutting-and-gluing procedure recover the weighted triangulation from the frieze.
Related
BibTeX
@misc{kuwana2026conwaycoxeter,
title = {{A Conway-Coxeter theorem for decorated frieze patterns}},
author = {Takeru Kuwana and Katsuhiko Matsuzaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06029},
eprint = {2609.06029},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06029},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Conway-Coxeter theorem for decorated frieze patterns
AU - Takeru Kuwana
AU - Katsuhiko Matsuzaki
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06029
UR - https://arxiv.org/abs/2609.06029
AB - Let m≥0 and put n=m+3. We introduce a normalized positive Laurent class of decorated frieze patterns and prove a decorated analogue of the Conway-Coxeter classification theorem. Namely, such friezes of width m are in canonical bijection with weighted triangulations of a convex n-gon, whose boundary edges are labeled by independent variables y_1,…,y_n and whose diagonals are labeled by x_1,…,x_m. While a weighted Conway-Coxeter propagation algorithm constructs the frieze from a weighted triangulation, a main new ingredient is an explicit nonrecursive Laurent formula expressing each quiddity entry directly from the weighted triangles incident to the corresponding vertex. Conversely, specialization to $1$, Laurent positivity, and a decorated cutting-and-gluing procedure recover the weighted triangulation from the frieze.
ER -