Guts of Surfaces and the Colored Jones Polynomial (Lecture Notes in Mathematics Book 2069)

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. Cover......Page 1 Guts of Surfaces and the Colored Jones Polynomial......Page 4 Preface......Page 6 Acknowledgements......Page 8 Contents......Page 10 1.1 History and Motivation......Page 12 1.2 State Graphs, and State Surfaces Far from Fibers......Page 15 1.3 Which Links are Semi-adequate?......Page 18 1.4 Essential Surfaces and Colored Jones Polynomials......Page 20 1.5 Volume Bounds from Topology and Combinatorics......Page 23 1.6 Organization......Page 24 2.1 State Circles and State Surfaces......Page 27 2.2 Decomposition into Topological Balls......Page 30 2.3 Primeness......Page 36 2.4 Generalizations to Other States......Page 40 3 Ideal Polyhedra......Page 44 3.1 Building Blocks of Shaded Faces......Page 45 3.2 Stairs and Arcs in Shaded Faces......Page 46 3.3 Bigons and Compression Disks......Page 54 3.4 Ideal Polyhedra for σ-Homogeneous Diagrams......Page 57 4.1 Maximal I-Bundles......Page 61 4.2 Normal Squares and Gluings......Page 64 4.3 Parabolically Compressing Normal Squares......Page 70 4.4 I-Bundles are Spanned by Essential Product Disks......Page 75 4.5 The σ-Adequate, σ-Homogeneous Setting......Page 79 5.1 Simple and Non-simple Disks......Page 81 5.2 Choosing a Spanning Set......Page 85 5.3 Detecting Fibers......Page 90 5.4 Computing the Guts......Page 91 5.5 Modifications of the Diagram......Page 94 5.6 The σ-Adequate, σ-Homogeneous Setting......Page 97 6.1 2-Edge Loops and Essential Product Disks......Page 99 6.2 Outline and First Step of Proof......Page 104 6.3 Step 2: Analysis Near Vertices......Page 105 6.4 Step 3: Building Staircases......Page 110 6.5 Step 4: Inside Non-prime Arcs......Page 115 7 Diagrams Without Non-prime Arcs......Page 117 7.1 Mapping EPDs to 2-Edge Loops......Page 118 7.2 A Four-to-One Mapping......Page 122 7.3 Estimating the Size of Ec......Page 125 8.1 Preliminaries......Page 127 8.2 Polyhedral Decomposition......Page 131 8.3 Two-Edge Loops and Essential Product Disks......Page 136 8.4 Excluding Complex Disks......Page 139 9.1 Volume Bounds for Hyperbolic Links......Page 147 9.2 Volumes of Montesinos Links......Page 152 9.3 Essential Surfaces and Colored Jones Polynomials......Page 157 9.4 Hyperbolic Volume and Colored Jones Polynomials......Page 160 10.1 Efficient Diagrams......Page 163 10.2 Control Over Surfaces......Page 165 10.4 A Coarse Volume Conjecture......Page 167 References......Page 170 Index......Page 174 1 Introduction 2 Decomposition into 3–balls 3 Ideal Polyhedra 4 I–bundles and essential product disks 5 Guts and fibers 6 Recognizing essential product disks 7 Diagrams without non-prime arcs 8 Montesinos links 9 Applications 10 Discussion and questions.