Guts of Surfaces and the Colored Jones Polynomial

This work initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A– or B–adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. 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, and 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 methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed. Our approach is to generalize the checkerboard decompositions of alternating knots and links. For A– or B–adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We use 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). In particular, we give a combinatorial formula for the complexity of the hyperbolic part of the JSJ decomposition (the guts) of the surface complement in terms of the diagram of the knot, and use this to give lower bounds on volumes of several classes of knots. Since state graphs have previously appeared in the study of Jones polynomials, our setting and methods create a bridge between quantum invariants and geometries of knot complements. D.F. affiliation: Department of Mathematics, Temple University, Philadelphia, PA 19122. Email: [email protected] E.K. affiliation: Department of Mathematics, Michigan State University, East Lansing, MI 48824. Email: [email protected] J.P. affiliation: Department of Mathematics, Brigham Young University, Provo, UT 84602. Email: [email protected]