Alphabeta MathColored Jones and the volume conjecture — Examples and Counterexamples
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Extensions, Extension Degree, and Finite Fields
- Applications of the Fundamental Group
- Approximation and Compactness in C(K)
- Artin Presentation Completeness and Braid Combing
- Asymptotic Cones and the Sublinear Triangle Criterion
- Binary Operations, Monoids, Groups and Subgroups
- Braided and Symmetric Monoidal Categories
- Braids as Fundamental Groups of Configuration Spaces
- Bruhat Decomposition and Flags over Finite Fields
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Cayley Graphs, Word Metrics and Quasi-Isometry
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Chains, Antichains, Sperner and Dilworth
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- Colored Jones and the volume conjecture
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Connectedness
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Crossing signs, writhe, and linking number
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Darboux, L'Hôpital, and Taylor's Theorem
- Derived Functors
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Duality and Rigidity in Monoidal Categories
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Exterior Powers, Orientation and Hodge Duality
- Fibrations Fiber Bundles and Homotopy Exact Sequences
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Foundations of the Real Numbers for Analysis
- Free Groups and Presentations
- Free Modules, Exact Sequences, Projective and Injective Modules
- Free Products and Amalgamation
- Fubini and Change of Variables
- Function Space Topologies and the Exponential Law
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Geodesics, the Exponential Map, Completeness, and Hopf–Rinow
- Geometric Braids and Artin Generators
- Graphs, Walks and Connectivity
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Hausdorff via the Diagonal
- Hecke Markov Traces and Polynomial Link Invariants
- Hereditary and Productive Behaviour of the Separation Axioms
- Higher Homotopy Groups and Cofiber Sequences
- Homotopy and Homotopy Equivalence
- Hurewicz Whitehead Freudenthal and Cw Approximation
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Kac Moody Algebras from Generalized Cartan Matrices
- Kauffman bracket state sum and the Jones polynomial
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Long Exact Sequences in Homology
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modular Categories, Verlinde Formula, and TQFT
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monoidal Categories and Monoidal Functors
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordered and Unordered Configuration Spaces
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Orientations Poincare Lefschetz and Alexander Duality
- Oriented Links, Braid Closures, and Markov Equivalence
- Partitions of Unity and Paracompactness
- Permutation Statistics, Inversions and Eulerian Numbers
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Plane Graphs, Euler's Formula and the Five Colour Theorem
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Principal Series Representations of GL N over a Finite Field
- Projective and Injective Resolutions
- Properties of the Integral and the Working FTC
- Punctured Disks, Mapping Classes, and Point Pushing
- Pure Braids, Fadell–Neuwirth, and Asphericity
- Quantized Enveloping Algebras and Quantum Serre Relations
- Rank Theorems and Embedded Submanifolds
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Ribbon categories and Reshetikhin–Turaev knot invariants
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sard Theorem and Transversality
- Semidirect Products, Automorphism Groups and Split Extensions
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Simply Connected Plane Domains: the Grand Equivalence
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Splitting Fields
- Strictification and Mac Lanes Coherence Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Temperley–Lieb Diagram Algebra and Catalan Basis
- Temperley–Lieb Root-of-Unity and Modular Representation Theory
- Temperley–Lieb Semisimplicity and Generic Structure
- Tensor and Fusion Categories
- Tensor Coherence and Algebraic Descent
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- The Artin Action on a Free Group
- The Ascoli–Arzelà Theorem
- The Burau Representations
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Diagram Lemmas in an Abelian Category
- The Exponential Function
- The Field of Fractions and Localisation
- The Fundamental Group
- The Fundamental Group of the Circle
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Trees, Forests and Spanning Trees
- Uniform Spaces: the Three Definitions
- Universal Coefficients and Kunneth Theorems
- Universal Properties, Representables and the Yoneda Lemma
- Urysohn's Lemma and the Tietze Extension Theorem
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Whitney Embedding Tubular Neighbourhoods and Approximation
- Yang–Baxter Operators and Quantum Braid Representations
- Young Diagrams Tableaux and Permutation Modules
2 · Logical flowchart
3 · Items
Counterexample
One finite sequence value does not determine asymptotic growth
Example
Unknot, trivial color, and a three-crossing braid value
Sources
- Hitoshi Murakami and Jun Murakami, The colored Jones polynomials and the simplicial volume of a knot, §5
- Hitoshi Murakami, An Introduction to the Volume Conjecture, §§1–4
- Hitoshi Murakami, An Introduction to the Volume Conjecture, §2.4 and Example 2.5
- Louis H. Kauffman, Combinatorial Knot Theory and the Jones Polynomial, §2