All Y-friezes come from $\mathrm{SL}_2$-friezes

Ian Short; Andrei Zabolotskii

All Y-friezes come from $\mathrm{SL}_2$-friezes

Combinatorics 2026-07-07 v1

Authors: Ian Short , Andrei Zabolotskii

Abstract

We resolve a conjecture of de Saint Germain: all Y-friezes arise from $\mathrm{SL}_2$ -friezes. Additionally, we give a threefold characterisation of those $\mathrm{SL}_2$ -friezes that give rise to the same Y-frieze: an elementary characterisation using quiddities, a combinatorial characterisation using triangulated polygons, and a geometrical characterisation using chains of horocycles.

Keywords

Y-friezes SL2-friezes quiddities triangulated polygons horocycles conjecture resolution

Cite

@article{arxiv.2607.06767,
  title  = {All Y-friezes come from $\mathrm{SL}_2$-friezes},
  author = {Ian Short and Andrei Zabolotskii},
  journal= {arXiv preprint arXiv:2607.06767},
  year   = {2026}
}

Comments

13 pages