Q-Deformed rationals and-continued fractions

Abstract

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