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arXiv · 2512.18810

$SL_2$-tilings with translational symmetry

Veronique Bazier-Matte·Marie-Anne Bourgie·Anna Felikson·Pavel Tumarkin
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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.

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Veronique Bazier-Matte, Marie-Anne Bourgie, Anna Felikson, Pavel Tumarkin. 2026-01-12. $SL_2$-tilings with translational symmetry. https://arxiv.org/abs/2512.18810

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