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.