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
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.
q-holonomic functions, D-modules, characteristic variety, deformation variety, colored Jones function, multisums, hypergeometric functions, WZ algorithm.
AMS subject classification.
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
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
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