Geodesic
$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS
Forum of Mathematics, Sigma, Tome 8 (2020)

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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. 
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     author = {SOPHIE MORIER-GENOUD and VALENTIN OVSIENKO},
     title = {$q${-DEFORMED} {RATIONALS} {AND} $q${-CONTINUED} {FRACTIONS}},
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SOPHIE MORIER-GENOUD; VALENTIN OVSIENKO. $q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.9

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