Geometry & Topology Monographs, Vol. 7 (2004),
Proceedings of the Casson Fest,
Paper no. 12, pages 291--309.

On the characteristic and deformation varieties of a knot

Stavros Garoufalidis

Abstract. The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). 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 C^2. We then compare it with the character variety of SL_2(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.

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

E-print: arXiv:math.GT/0306230

Submitted to GT on 16 June 2003. (Revised 1 November 2003.) Paper accepted 15 December 2003. Paper published 20 September 2004.

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Stavros Garoufalidis
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA

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