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Crossing signs, writhe, and linking number — Examples and Counterexamples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- 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
- Crossing signs, writhe, and linking number
- Darboux, L'Hôpital, and Taylor's Theorem
- 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
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Formal Immersions and the Smale Hirsch Theorem
- Foundations of the Real Numbers for Analysis
- 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
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Isotopy Extension and Embedding Theory Beyond Whitney
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- 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
- 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
- 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
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- Relations, Functions, and Quotients
- 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
- 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
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- 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
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- 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 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
- 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
Hopf and Whitehead link linking-number computations
Example
Assume . Read a braid word in the standard geometric braid group product convention: the rightmost factor is traversed first from bottom to top, and the closure joins corresponding top and bottom endpoints. A positive Artin half twist has crossing sign .
The closure of the positive two-strand braid is the positive Hopf link. Its endpoint permutation is the identity, so it has two components, and both crossings are mixed positive crossings. Hence its linking number is .
The standard Whitehead link is the closure of the three-strand braid Its endpoint permutation is , so its components are the cycles and . In bottom-to-top traversal order the crossing letters are . The first, second, third, and fifth crossings are mixed, with signs ; the fourth is a self-crossing. The mixed sum is , so the linking number is . The five-crossing braid diagram is a representative, not a claim that the displayed braid word is the unique presentation.
Example references
Diagram
The two-strand braid in the figure is read upward. At each positive generator crossing, the left-to-right strand passes under and is interrupted by the grey-white gap; the right-to-left strand passes over. Between the crossings, label is on the left and label on the right. The outside arcs close the top endpoints to the matching bottom endpoints.
Facts & Assumptions
Given: , the two displayed braid words, and the crossing-sign convention of this pair.
The Axiom of Countable Choice, written (The Axiom of Countable Choice ()).
The endpoint permutation of a geometric braid records the endpoint of each labelled strand, and the components of its closure are exactly the cycles of that permutation (Geometric braids in the disc with setwise endpoints, The closure of a geometric braid).
Stacking induces the braid-group product, with endpoint permutations multiplying in the same product order (Stacking of geometric braids is a well-defined associative operation on isotopy classes).
The positive geometric half twist is the generator and is calibrated to crossing sign ; its inverse has the opposite crossing sign (The elementary geometric half twist, its support disc, and its opposite, Oriented crossing signs, diagram writhe, and blackboard framing).
For an accepted two-component diagram, the algorithm returns half the signed sum of its mixed crossings (Compute oriented linking number from a diagram, The diagram linking-number scan terminates and returns the stated integer).
Verification
For the two-strand braid, by [F1] and [F2]. Its two closure cycles are therefore the two singleton components, and each of the two positive generator crossings is mixed. By [F3] their signs are ; [F4] gives linking number .
For the Whitehead word, each squared generator contributes the identity endpoint permutation, leaving ; hence [F1] gives component cycles and . Traversing from bottom to top, the first two crossings exchange labels and , so they are mixed; the following exchanges and , so it is mixed; the next exchanges and , so it is a self-crossing; the final exchanges and , so it is mixed.
By [F3], the mixed signs of the Whitehead word are in that traversal, whose sum is . The algorithm in [F4] therefore returns , as claimed. These are finite crossing checks; the general invariance proof is separate.
Writhe is not an invariant of an unframed knot
Statement refuted
The false statement is: for every regular oriented diagram of an unframed knot, the writhe , defined as the sum of its oriented crossing signs, depends only on the ambient-isotopy class of the represented knot.
Statement references
Diagram
The right-hand curve is one oriented circle with a single local kink. At the over tangent is and the under tangent is in the positive projection chart, so its crossing sign is . The lower-right loop is the added type-I kink; deleting it gives the crossing-free unknot diagram on the left. The figure illustrates the construction; the proof uses the Reidemeister theorem and the signed-sum definition.
Facts & Assumptions
Given: The Axiom of Countable Choice and the false statement above.
The Axiom of Countable Choice, written (The Axiom of Countable Choice ()).
A regular oriented diagram represents a smooth oriented link, and an oriented Reidemeister type-I move adds or removes a self-crossing kink (Regular oriented link diagrams, Oriented Reidemeister moves).
Under , two regular diagrams connected by an oriented Reidemeister move represent ambient-isotopic oriented links (Reidemeister's theorem for oriented diagrams).
The writhe is the sum of crossing signs; a positive type-I crossing contributes (Oriented crossing signs, diagram writhe, and blackboard framing).
Blackboard self-linking equals diagram writhe (Blackboard self-linking equals diagram self-writhe).
Counterexample
Let be a crossing-free diagram of the oriented unknot and let be obtained by one positive type-I move. By [F1] the new crossing is a self-crossing and is positive.
By [F2], the type-I move is realized by an ambient isotopy, so and represent the same unframed oriented knot. By [F3], while . Therefore the displayed false statement fails.
By [F4], the blackboard self-linking values for these two diagrams are also and , respectively. This is the framing change carried by the single positive kink; it does not contradict equality of knot types after forgetting the framing.