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Algebraic concordance and metabolic forms — Examples and Counterexamples
1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic concordance and metabolic forms
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- 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
- Chern and Pontryagin Classes by Splitting and Complexification
- Classification of Compact Connected Surfaces
- Classification of Covering Spaces
- Codimension One Foliations, Secondary Classes and Characteristic Disk Foundations
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- 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
- Cyclic Groups and Direct Products
- 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
- 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
- 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
- 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
- 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
- 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
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Orientations Poincare Lefschetz and Alexander Duality
- Oriented and Mod Two Intersection Numbers
- Oriented Links, Braid Closures, and Markov Equivalence
- Partitions of Unity and Paracompactness
- 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
- Projective and Injective Resolutions
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Reeb Stability and Global Foliation Constructions
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Relative Homology Excision and Mayer Vietoris
- Riemannian Metrics Length Distance and Volume
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sard Theorem and Transversality
- Seifert matrices and the Alexander module
- Seifert surfaces and knot genus
- 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 and topological knot concordance
- 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 de Rham Theorem and Degree
- 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
- Trees, Forests and Spanning Trees
- Universal Coefficients and Kunneth Theorems
- Universal Properties, Representables and the Yoneda Lemma
- Vector Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Whitney Embedding Tubular Neighbourhoods and Approximation
2 · Logical flowchart
3 · Definitions, theorems and proofs
None yet.
4 · Examples, counterexamples and false statements
Worked cases for metabolic Seifert forms
Example
Assume full AC. The disk Seifert surface of the oriented unknot has the rank-zero Seifert lattice and is metabolic. In the positive-push convention, the standard positive Hopf-band plumbing for the right-handed trefoil has matrix . Since is invertible, its rational isometric structure has symmetric form , which is negative definite; the integral Seifert lattice is not metabolic.
Example references
Facts & Assumptions
Given: Full AC, the disk Seifert surface of the oriented unknot, and the standard positive Hopf-plumbing Seifert surface of the right-handed trefoil.
Full AC is the choice-function axiom used by the knot Seifert-pairing interface. The Axiom of Choice
The rank-zero lattice is allowed; its zero subgroup is a primitive rank-zero metabolizer. Metabolic Seifert forms and algebraic knot concordance
The positive Hopf-plumbing basis has matrix . A positive Hopf plumbing gives the trefoil Seifert matrix
Since is invertible, satisfies the rational isometric-structure definition. Integral Witt classes and rational Seifert isometric structures, An invertible Seifert matrix gives a rational isometric structure
A metabolic lattice must have a rank-half direct summand on which the Seifert pairing vanishes. Metabolic Seifert forms and algebraic knot concordance
Proof
The disk has , so its Seifert matrix is the empty matrix. The subgroup has rank zero, is a direct summand, and the pairing vanishes on it. By [F1], this is metabolic and represents the identity class.
By [F2] and [F3], the trefoil's rational structure has symmetric form . For a vector , , which is negative whenever . Thus the rational structure has no nonzero isotropic line. For an integral vector the Seifert self-pairing is , so no rank-one subgroup has identically zero Seifert pairing, and [F4] rules out an integral metabolizer.
Unimodular skew part does not force metabolicity
Statement refuted
Assume full AC. The false claim is: every positive-rank integral Seifert lattice with unimodular alternating part is metabolic.
Statement references
Facts & Assumptions
Given: Full AC and the positive-push Seifert matrix of the right-handed trefoil.
Full AC is the choice-function axiom used by the knot Seifert-pairing interface. The Axiom of Choice
The positive Hopf plumbing has matrix . A positive Hopf plumbing gives the trefoil Seifert matrix
A metabolic rank-two lattice must have a rank-one subgroup on which vanishes, and hence every vector in that subgroup has zero self-pairing. Metabolic Seifert forms and algebraic knot concordance
Since is invertible, satisfies the rational isometric-structure definition. Integral Witt classes and rational Seifert isometric structures, An invertible Seifert matrix gives a rational isometric structure
A rational isotropic line for an integral bilinear form saturates to a rank-one integral direct summand with zero restriction. A rational isotropic subspace saturates to an integral direct summand
Counterexample
The right-handed trefoil's positive-push Seifert lattice has and . Its symmetric form is negative definite, so the lattice is not metabolic, as the calculation below proves.
From [F1], has determinant , so this is a positive-rank integral Seifert lattice satisfying the false claim's hypothesis.
By [F3], the associated rational form is . For every nonzero , , so there is no rational isotropic line to which [F4] applies. In particular, for every nonzero integral vector the Seifert self-pairing is negative; [F2] rules out a rank-one metabolizer. This lattice refutes the claim.