Irreducibility of q–difference operators and the knot 74

Garoufalidis, Stavros; Koutschan, Christoph

doi:10.2140/agt.2013.13.3261

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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

DOI: 10.2140/agt.2013.13.3261

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 $SL (2, ℂ)$ 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.

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

References

Publication

Received: 26 November 2012

Accepted: 30 April 2013

Published: 10 October 2013

Authors

Stavros Garoufalidis
School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160 USA
http://www.math.gatech.edu/~stavros
Christoph Koutschan
Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences Altenberger Straße 69 A-4040 Linz Austria
http://www.koutschan.de

Volume 13, issue 6 (2013)