#### Volume 13, issue 6 (2013)

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

### Stavros Garoufalidis and Christoph Koutschan

Algebraic & Geometric Topology 13 (2013) 3261–3286
##### 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.

##### Keywords
$q$–holonomic module, $q$–holonomic sequence, creative telescoping, irreducibility of $q$–difference operators, factorization of $q$–difference operators, qHyper, Adams operations, quantum topology, knot theory, colored Jones polynomial, AJ conjecture, double twist knot, $7_4$
##### Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57M25, 33F10, 39A13