Irreducibility of $q$–difference operators and the knot $7_4$

Abstract

Our goal is to compute the minimal-order recurrence of the colored Jones polynomial of the $7_4$ knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot $7_4$ with reducible nonabelian $SL\left(2,ℂ\right)$ character variety. To achieve our goal, we use symbolic summation techniques of Zeilberger’s holonomic systems approach and an irreducibility criterion for $q$–difference operators. For the latter we use an improved version of the qHyper algorithm of Abramov–Paule–Petkovšek to show that a given $q$–difference operator has no linear right factors. En route, we introduce exterior power Adams operations on the ring of bivariate polynomials and on the corresponding affine curves.