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$q$-rationals and dimers

Valentin Ovsienko

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.

$q$-rationals and dimers

Abstract

We describe the relationships between the notion of -deformed rational numbers, introduced in our previous work with Sophie Morier-Genoud, and the theory of dimer models. We show that -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 -rationals.
Paper Structure (21 sections, 4 theorems, 49 equations, 11 figures)

This paper contains 21 sections, 4 theorems, 49 equations, 11 figures.

Table of Contents

  1. Introduction and the main result
  2. A brief introduction to $q$-rationals
  3. The modular group $\mathrm{PSL}(2,\mathbb{Z})$
  4. $q$-deformed continued fractions
  5. Continued fractions in the matrix form
  6. Continuants
  7. An example: the $q$-Fibonacci sequence
  8. Rational numbers and snake graphs
  9. Definition of the snake graph $\mathcal{G}_{\frac{r}{s}}$
  10. The ladders
  11. Proof of the main theorem
  12. Detailed statement of the main theorem
  13. Two inductive cases
  14. Case $1$
  15. Case $2$
  16. ...and 6 more sections

Key Result

Theorem 1.3

For every rational $\frac{r}{s}\geq1$, and the corresponding $q$-deformed rational $\left[\frac{r}{s}\right]_q=\frac{\mathcal{R}(q)}{\mathcal{S}(q)}$, the polynomial $\mathcal{R}$ in the numerator is equal (up to a scalar multiple) to the weighted number of perfect matchings $\mathcal{M}_q(\mathcal{

Figures (11)

  • Figure 4: Elementary boxes.
  • Figure 5: Two sign arrangements on the initial box.
  • Figure 6:
  • Figure 7: a) the snake for $8$; b) attaching two ladders.
  • Figure 8: Attachment of a box to $\mathcal{G}_{\frac{r'}{s'}}$: Case $1$.
  • ...and 6 more figures

Theorems & Definitions (24)

  • Definition 1.1
  • Remark
  • Definition 1.2
  • Remark
  • Theorem 1.3
  • Remark
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Remark
  • ...and 14 more