  In this paper, we study the desingularization problem in the first $q$-Weyl
algebra. We give an order bound for desingularized operators, and thus derive
an algorithm for computing desingularized operators in the first $q$-Weyl
algebra. Moreover, an algorithm is presented for computing a generating set of
the first $q$-Weyl closure of a given $q$-difference operator. As an
application, we certify that several instances of the colored Jones polynomial
are Laurent polynomial sequences by computing the corresponding desingularized
operator.
