Geometry & Topology Monographs 7 (2004) 291–309

DOI: 10.2140/gtm.2004.7.291

Abstract

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose nth term is the Jones polynomial of the knot colored with the n–dimensional irreducible representation of sl₂. It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q–holonomic, ie, that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1–dimensional variety in C². We then compare it with the character variety of SL₂(C) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots.

We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the so-called noncommutative A–polynomial) of the characteristic variety of a knot.

Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter.

Keywords

q–holonomic functions, D–modules, characteristic variety, deformation variety, colored Jones function, multisums, hypergeometric functions, WZ algorithm.

Mathematical Subject Classification

Primary: 57N10

Secondary: 57M25

Publication

Received: 16 June 2003
Revised: 1 November 2003
Accepted: 15 December 2003
Published: 20 September 2004

Authors