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Our goal is to compute the minimal-order recurrence of the colored Jonespolynomial 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 withreducible non-abelian SL(2,C) character variety. To achieve our goal, we usesymbolic summation techniques of Zeilberger's holonomic systems approach and anirreducibility criterion for q-difference operators. For the latter we use animproved version of the qHyper algorithm of Abramov-Paule-Petkovsek to showthat a given q-difference operator has no linear right factors. En route, weintroduce exterior power Adams operations on the ring of bivariate polynomialsand on the corresponding affine curves.

Irreducibility of q-difference operators and the knot 7_4