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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

Our goal is to compute the minimal-order recurrence of the colored Jones polynomial 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 nonabelian SL(2, ) 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.

$q$–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
Received: 26 November 2012
Accepted: 30 April 2013
Published: 10 October 2013
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
686 Cherry Street
Atlanta, GA 30332-0160
Christoph Koutschan
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Austrian Academy of Sciences
Altenberger Straße 69
A-4040 Linz