q-deformed integers derived from pairs of coprime integers and its applications

Abstract

In connection with cluster algebras, snake graphs and q-integers, Kyungyong Lee and Ralf Schiffler recently found a formula for computing the (normalized) Jones polynomials of rational links in terms of continued fraction expansion of rational numbers. Sophie Morier-Genoud and Valentin Ovsienko introduced q-deformed continued fractions, and showed that by using them each coefficient of the normalized Jones polynomial counted quiver representations of type A_n. In this paper we introduce q-deformed integers defined by pairs of coprime integers, which are motivated by the denominators and the numerators of their q-deformed continued fractions, and give an efficient algorithm for computing the (normalized) Jones polynomials of rational links. Various properties of q-integers defined by pairs of coprime integers are investigated and shown its applications.