$SL_2$-tilings with translational symmetry

Veronique Bazier-Matte; Marie-Anne Bourgie; Anna Felikson; Pavel Tumarkin

$SL_2$-tilings with translational symmetry

Combinatorics 2026-01-13 v2 Rings and Algebras

Authors: Veronique Bazier-Matte , Marie-Anne Bourgie , Anna Felikson , Pavel Tumarkin

Abstract

An $SL_2$ -tiling is a bi-infinite matrix in which all adjacent $2 \times 2$ minors are equal to $1$ . Positive integral $SL_2$ -tilings were introduced by Assem, Reutenauer and Smith as generalisations of classical Conway--Coxeter frieze patterns. We show that positive integral $SL_2$ -tilings with translational symmetry are in bijection with triangulations of annuli. We use this correspondence to study the properties of periodic positive integral $SL_2$ -tilings.

Keywords

SL2 tilings frieze patterns triangulations translational symmetry combinatorial enumeration

Cite

@article{arxiv.2512.18810,
  title  = {$SL_2$-tilings with translational symmetry},
  author = {Veronique Bazier-Matte and Marie-Anne Bourgie and Anna Felikson and Pavel Tumarkin},
  journal= {arXiv preprint arXiv:2512.18810},
  year   = {2026}
}

Comments

15 pages, many figures; v2: another way to obtain the main results is explained, this was communicated to the authors by Peter Jorgensen

$SL_2$-tilings with translational symmetry

Abstract

Keywords

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