Volume 9, issue 3 (2005)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The colored Jones function is q-holonomic

Stavros Garoufalidis and Thang T Q Le

Geometry & Topology 9 (2005) 1253–1293

arXiv: math.GT/0309214

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
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
References
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Publication
Received: 28 October 2004
Revised: 20 July 2005
Accepted: 3 July 2005
Published: 24 July 2005
Proposed: Walter Neumann
Seconded: Joan Birman, Vaughan Jones
Authors
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
Atlanta
Georgia 30332-0160
USA
http://www.math.gatech.edu/~stavros/
Thang T Q Le
School of Mathematics
Georgia Institute of Technology
Atlanta
Georgia 30332-0160
USA