$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS

By Sophie Morier-Genoud, Valentin Ovsienko. Forum of Mathematics, Sigma, 2018

82 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ccd355b9c70b8b2546a2c357b6ab3e7fa6e6f670

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In Computational Theory and Mathematics

1. Sophie Morier-Genoud
2. Valentin Ovsienko

In Mathematical Physics

1. Sophie Morier-Genoud
2. Valentin Ovsienko

Citations in the corpus

• Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections (2018, background): “One application is already explored in the second part of this work [28], where we suggest a notion of q -deformed continued fractions and of q -deformed rational numbers; this deformation preserves the combinatorial properties discussed in the present paper.”