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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.
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