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

Part of: Elementary number theory Low-dimensional topology Sequences and sets Combinatorics Arithmetic rings and other special rings

Published online by Cambridge University Press:  06 March 2020

SOPHIE MORIER-GENOUD  and
VALENTIN OVSIENKO Open the ORCID record for VALENTIN OVSIENKO [Opens in a new window]
SOPHIE MORIER-GENOUD
Affiliation:
Sorbonne Université, Université Paris Diderot, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75005, Paris, France; [email protected]
VALENTIN OVSIENKO
Affiliation:
Centre national de la recherche scientifique, Laboratoire de Mathématiques, UMR du CNRS 9008, U.F.R. Sciences Exactes et Naturelles Moulin de la Housse - BP 1039 51687 Reims cedex 2, France; [email protected]

Abstract

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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.

MSC classification

Primary: 05A30: $q$-calculus and related topics 11A55: Continued fractions 11B57: Farey sequences; the sequences ${1^k, 2^k, cdots}$ 13F60: Cluster algebras 57M27: Invariants of knots and 3-manifolds
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e13
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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