Irreducibility of q-difference operators and the knot 7_4

Our goal is to compute the minimal-order recurrence of the colored Jones poly- nomial of the 74 knot, as well as for the first four double twist knots. As a corollary, we verify the AJ Conjecture for the simplest knot 74 with reducible non-abelian SL(2, ) char- acter 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-Petkovysek to show that a given q-difference operator has no linear right factors. En route, we in- troduce exterior power Adams operations on the ring of bivariate polynomials and on the corresponding affine curves.

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