  The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$,
with integer coefficients. Many people have pondered why is this so, and what
is a proper generalization of the Jones polynomial for knots in other closed
3-manifolds. Our paper centers around this question. After reviewing several
existing definitions of the Jones polynomial, we show that the Jones polynomial
is really an analytic function, in the sense of Habiro. Using this, we extend
the holonomicity properties of the colored Jones function of a knot in 3-space
to the case of a knot in an integer homology sphere, and we formulate an
analogue of the AJ Conjecture. Our main tools are various integrality
properties of topological quantum field theory invariants of links in
3-manifolds, manifested in Habiro's work on the colored Jones function.Revised
version updating references on Witten-Reshetikhin-Turaev invariants.
