  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.
