  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 that arise naturally in
the study of knot polynomial invariants (A- or B-adequacy), 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 in the knot
complement; in particular, the surface is a fiber if and only if a particular
coefficient vanishes. Our results also yield concrete relations between
hyperbolic geometry and colored Jones polynomials: for certain families of
links, coefficients of the polynomials determine the hyperbolic volume to
within a factor of 4.
  Our approach is to generalize the checkerboard decompositions of alternating
knots. Under mild diagrammatic hypotheses (A- or B-adequacy), we show that the
checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral
decomposition of their complement. We employ normal surface theory to establish
a dictionary between the pieces of the JSJ decomposition of the surface
complement and the combinatorial structure of certain spines of the
checkerboard surface (state graphs). Since state graphs have previously
appeared in the study of Jones polynomials, our setting and methods create a
bridge between quantum and geometric knot invariants.
