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