  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.
