-Deformed Rationals and -Continued Fractions

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.