This monograph derives direct and concrete relations between colored
Jones polynomials and the topology of incompressible spanning surfaces
in knot and link complements. Under mild diagrammatic hypotheses, we
prove that the growth of the degree of the colored Jones polynomials is
a boundary slope of an essential surface in the knot complement. We show
that certain coefficients of the polynomial measure how far this surface
is from being a fiber for the knot; in particular, the surface is a
fiber if and only if a particular coefficient vanishes. We also relate
hyperbolic volume to colored Jones polynomials. Our method is to
generalize the checkerboard decompositions of alternating knots. Under
mild diagrammatic hypotheses, we show that these surfaces are essential,
and obtain an ideal polyhedral decomposition of their complement. We use
normal surface theory to relate the pieces of the JSJ decomposition of
the complement to the combinatorics of certain surface spines (state
graphs). Since state graphs have previously appeared in the study of
Jones polynomials, our method bridges the gap between quantum and
geometric knot invariants.