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.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
25 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献