By David Futer
This monograph derives direct and urban relatives among coloured Jones polynomials and the topology of incompressible spanning surfaces in knot and hyperlink enhances. below gentle diagrammatic hypotheses, we turn out that the expansion of the measure of the coloured Jones polynomials is a boundary slope of a vital floor within the knot supplement. We express that sure coefficients of the polynomial degree how a long way this floor is from being a fiber for the knot; specifically, the outside is a fiber if and provided that a specific coefficient vanishes. We additionally relate hyperbolic quantity to coloured Jones polynomials. Our technique is to generalize the checkerboard decompositions of alternating knots. below light diagrammatic hypotheses, we express that those surfaces are crucial, and procure a fantastic polyhedral decomposition in their supplement. We use basic floor idea to narrate the items of the JSJ decomposition of the supplement to the combinatorics of sure floor spines (state graphs). because country graphs have formerly seemed within the learn of Jones polynomials, our strategy bridges the space among quantum and geometric knot invariants.