$q$-DEFORMED RATIONALS AND $q$-CONTINUED FRACTIONS. (6th March 2020)

• Record Type: Journal Article
• Authors: MORIER-GENOUD, SOPHIE; OVSIENKO, VALENTIN
• Is Part Of: Forum of mathematics. Volume 8(2020)
• Publication Date: 2020-03-06
• Subjects: 05A30, -- 11A55, -- 11B57, -- 13F60, -- 57M27; Mathematics -- Periodicals; 510
• DOI: 10.1017/fms.2020.9
• Languages: English
• ISSNs: 2050-5094

Abstract

Abstract : 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.