q-DEFORMED RATIONALS AND q-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 identitiy for the Gaussian binomial coefficients, but...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2020, Vol.8
Main Authors: Morier-Genoud, Sophie, Ovsienko, Valentin
Format: Article
Language:English
Subjects:
05A30
11A55
11B57
13F60
57M27
Mathematics
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Summary: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 identitiy 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.
ISSN:2050-5094
DOI:10.1017/fms.2020.9