EAN: 9783642333019

GUTS OF SURFACES AND THE COLORED JONES POLYNOMI...

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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