#### 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.

Notes on file formats
Stavros Garoufalidis

School of Mathematics, Georgia Institute of Technology

Atlanta, GA 30332-0160, USA

Email: stavros@math.gatech.edu

URL: http://www.math.gatech.edu/~stavros/

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