Geometry & Topology 9 (2005) 1253–1293

DOI: 10.2140/gt.2005.9.1253

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

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