Alphabeta MathHow statement and proof provenance work
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Pairwise linking numbers do not classify labeled links
Statement refuted
The matrix of pairwise linking numbers determines every labeled oriented link in up to orientation-preserving ambient isotopy.
Statement references
Facts & Assumptions
Given: The Borromean and unlink diagrams with the labels and orientations in the counterexample statement.
Full AC, written AC, is The Axiom of Choice.
The Borromean closure has all pairwise linking numbers zero and its based triple coefficient is . The Borromean string link has nonzero triple Milnor invariant
Every component of the three-component unlink bounds a disk disjoint from the other components, so its pairwise linking numbers and its based longitude words are zero and trivial, respectively. Based string links and labeled link peripheral systems
Closed-link Milnor residues are invariant under labeled ambient isotopy. Isotopy invariance and nonrepeating link-homotopy scope of Milnor invariants
Counterexample
Compare the three-component unlink with the closure of the based Borromean string link in the preceding example. Every pairwise linking number is zero for both links, while the based triple Milnor residue candidate is for the unlink and for the Borromean closure.
Given: The two labeled oriented links in the counterexample statement.
By [F1] and [F2], both links have the zero pairwise linking matrix. The unlink longitude words are trivial, so its based triple residue candidate is , whereas the Borromean based candidate is .
Under [A1], if the two links were orientation-preservingly ambient isotopic as labeled links, [F3] would force their triple Milnor residue candidates to agree. Step 1.1 gives distinct candidates because the pairwise-linking indeterminacy is zero. The modulus is zero for both links, so this is an inequality of integers, and [F3] applies without a basing qualification. This contradiction proves that the links are not ambient isotopic and refutes the asserted classification.
Depends on
- The Axiom of Choice
- Based string links and labeled link peripheral systems
- The Borromean string link has nonzero triple Milnor invariant
- Isotopy invariance and nonrepeating link-homotopy scope of Milnor invariants
- Compute oriented linking number from a diagram
- The diagram linking-number scan terminates and returns the stated integer
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
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Sources
- John Milnor, Link Groups (standard reference, not scraped)
- Ryan Stees, Milnor’s Invariants for Knots and Links in Closed Orientable 3-Manifolds (standard reference, not scraped)