We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

中文翻译：

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- 变形有理数和 - 连续分数

我们引入一个概念 $q$ -变形有理数和 $q$ -变形的连分数。一种 $q$ -变形有理由多边形的三角剖分编码，并且可以递归计算。递归公式类似于 $q$ -高斯二项式系数的变形帕斯卡恒等式，但帕斯卡三角形被 Farey 图取代。多项式的系数定义 $q$ - 有理数颤动图的最大不可分解表示的子表示对三角剖分。其他几个属性，例如总正性属性， $q$ -Farey 图的变形，矩阵表示和 $q$ 给出了 - 连续数，以及与有理结的琼斯多项式的关系。