Alphabeta MathMultivariable Alexander modules and chain models
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Applications of the Fundamental Group
- Binary Operations, Monoids, Groups and Subgroups
- Cardinal Arithmetic, Cofinality and the Alephs
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Chain Homotopy and the Homotopy Category
- Chern and Pontryagin Classes by Splitting and Complexification
- Classification of Covering Spaces
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- 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
- 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
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exactness and the Member Calculus
- Ext and Balanced Resolutions
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- 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
- Geometric Braids and Artin Generators
- 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
- Homology Axioms Degree and Classical Applications
- 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
- Knot exteriors and peripheral systems
- 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
- Local Coefficients, Twisted Homology, and Duality
- Localisation of Modules and Support
- 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
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Noetherian Rings and Hilbert Basis
- 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
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- 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
- Projective and Injective Resolutions
- Properties of the Integral and the Working FTC
- Pure Braids, Fadell–Neuwirth, and Asphericity
- Quasi Coherent and Coherent Sheaves and Vector Bundles
- Rank Theorems and Embedded Submanifolds
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Relative smooth triangulation and bundle fitting
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sard Theorem and Transversality
- 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
- 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
- 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
- Tensor Fields Exterior Algebra and Differential Forms
- Tensor Products of Modules
- 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 Group Algebra and Representations of Finite Groups
- 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
- Universal Coefficients and Kunneth Theorems
- Universal Properties, Representables and the Yoneda Lemma
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Whitney Embedding Tubular Neighbourhoods and Approximation
2 · Logical flowchart
3 · Items
Lemma
The integral multivariable Laurent polynomial ring is Noetherian and a UFD
Definition
Based string links and labeled link peripheral systems
Definition
The maximal-abelian-cover Alexander module of an oriented link
Lemma
A punctured-sphere cover from a labeled link diagram
Lemma
The maximal abelian cover retracts onto the cover over the compact link exterior
Lemma
String-link exterior homology and end-meridian maps
Lemma
Local meridian conjugacy at a labeled link crossing
Definition
Topological linking number as a meridian coordinate
Lemma
Cellular chains of the maximal abelian cover compute the link Alexander module
Lemma
Diagram word for a preferred peripheral longitude
Algorithm
Build the maximal-abelian-cover chain complex from a marked triangulated exterior
Lemma
Fitting ideals, order, and diagonal specialization of the multivariable Alexander module
Theorem
Termination and exact validation of the triangulated-exterior chain procedure
Theorem
The triangulated-exterior chain model computes the multivariable Alexander module
Sources
- Keith Conrad, Eisenstein Criterion and Gauss' Lemma, Theorem 1.3
- The Stacks Project, Section 10.31 (Tag 00FM), Noetherian rings
- The Stacks Project, Section 10.9 (Tag 00CM), Localization
- Cohen, Komendarczyk, Koytcheff and Shonkwiler, Homotopy String Links and the kappa-Invariant
- John Milnor, Link Groups
- Ryan Stees, Milnor’s Invariants for Knots and Links in Closed Orientable 3-Manifolds
- Daniel S. Silver and Susan G. Williams, Mahler Measure, Links and Homology Growth, §4, pp. 4–5
- Allen Hatcher, Algebraic Topology, §1.3
- Thomas Cochran, Classical Isotopy Invariants of Links and 3-Manifolds, slides 4–5
- Stefan Friedl, Knot Theory, §5.3, proof of the Wirtinger Presentation Proposition
- Allen Hatcher, Algebraic Topology, §1.3, homotopy lifting and covering classification
- J. Peter May, A Concise Course in Algebraic Topology, Chapter 3
- Cole Charbonneau, Computing the Wirtinger Presentation of a Knot Group from a Knot Diagram
- Allen Hatcher, Algebraic Topology, §2.2, Theorem 2.35, printed pp. 139–140
- Stefan Friedl, Knot Theory, §7.1, Longitudes
- Mohamed Elhamdadi, Mustafa Hajij, and Kyle Istvan, Framed Knots
- Allen Hatcher, Algebraic Topology, §2.2, Theorem 2.35
- The Stacks Project, Section 15.8 (Tag 07Z6), Fitting ideals
- Daniel S. Silver and Susan G. Williams, Mahler Measure, Links and Homology Growth, §4
- Allen Hatcher, Algebraic Topology, §2.2