Guts of surfaces and the colored Jones polynomial

TL;DR

This paper establishes concrete links between colored Jones polynomials and the topology of surfaces in knot complements, revealing how polynomial coefficients relate to geometric structures and hyperbolic volume.

Contribution

It introduces a novel approach connecting quantum invariants with geometric topology using checkerboard decompositions and normal surface theory under diagrammatic hypotheses.

Findings

Degree growth of polynomials corresponds to boundary slopes of surfaces.

Certain polynomial coefficients indicate whether a surface is a fiber.

Coefficients can approximate hyperbolic volume within a factor of 4.

Abstract

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.