Geometry & Topology, Vol. 9 (2005) Paper no. 29, pages 1253--1293.

The colored Jones function is q-holonomic

Stavros Garoufalidis and Thang TQ Le

Abstract. A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q-proper-hypergeometric function, and thus it is q-holonomic. We demonstrate our results by computer calculations.

Keywords. Holonomic functions, Jones polynomial, Knots, WZ algorithm, quantum invariants, D-modules, multisums, hypergeometric functions

AMS subject classification. Primary: 57N10. Secondary: 57M25.

E-print: arXiv:math.GT/0309214

DOI: 10.2140/gt.2005.9.1253

Submitted to GT on 28 October 2004. (Revised 20 July 2005.) Paper accepted 3 July 2005. Paper published 24 July 2005.

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Stavros Garoufalidis Thang TQ Le
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
Email: stavros@math.gatech.edu, letu@math.gatech.edu
URL: http://www.math.gatech.edu/~stavros

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