Irreducibility of q–difference operators and the knot 74

Garoufalidis, Stavros¹ ; Koutschan, Christoph²

¹School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA
²Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040 Linz, Austria

Algebraic and Geometric Topology, Tome 13 (2013) no. 6, pp. 3261-3286

Résumé

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, ℂ) 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.

Détail

DOI : 10.2140/agt.2013.13.3261

Classification : 57N10, 57M25, 33F10, 39A13
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, $7_4$

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