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arXiv · 2510.16270

$q$-rationals and dimers

Valentin Ovsienko
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Abstract

We describe the relationships between the notion of $q$-deformed rational numbers, introduced in our previous work with Sophie Morier-Genoud, and the theory of dimer models. We show that $q$-deformed rationals can be calculated in terms of perfect matchings of certain bipartite graphs, known as snake graphs, or ribbon tiles, etc. equipped with a certain weight function on the set of edges. We apply some elements of the dimer theory to get more information about $q$-rationals.

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Valentin Ovsienko. 2025-10-17. $q$-rationals and dimers. https://arxiv.org/abs/2510.16270

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