Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The classical signature bounds unknotting number

Statement

Assume full AC. For every smooth oriented knot K, the classical signature σ−1(K) of the positive-push Seifert form is independent of the chosen connected spanning surface and integral basis, and it satisfies ∣σ−1(K)∣≤2u(K). Zero eigenvalues are omitted when computing the signature of a Hermitian matrix.

Facts & Assumptions

Given: Full AC and a smooth oriented knot K.

[A1]

Full AC implies the countable-choice hypothesis of the unknotting-number definition. The Axiom of Choice, AC implies DC implies countable choice

[F1]

The unknotting set is nonempty and u(K) is its least element, measured by finite sequences of crossing changes in regular diagrams. Bridge number and unknotting number of an oriented knot

[F2]

A single crossing change admits corresponding Seifert matrices with difference zero or ±vvT. One crossing change changes a Seifert matrix by rank at most one

[F3]

Two Hermitian matrices differing by rank at most one have signatures differing by at most two, with zero eigenvalues omitted. A rank-one Hermitian perturbation changes signature by at most two

[F4]

The positive-push matrix gives H−1(V)=2(V+VT); surface and basis changes preserve its signature. At a nonroot this is the declared invariant σ−1. Levine–Tristram Hermitian form and signature, Seifert stabilization preserves the Levine–Tristram signature, The singular parameters of the Levine–Tristram form

[F5]

The skew part V−VT is the unimodular intersection pairing of a knot Seifert surface. The skew part of a Seifert matrix is the surface intersection form

Proof

1.1A1F4F5givenalgebra

At ω=−1 the form is nonsingular for every knot. Indeed V+VT≡V−VT(mod2), so its determinant is odd by [F5] and in particular is not zero. Thus H−1=2(V+VT) is nonsingular. By [F4], −1 is not an Alexander root and the actual matrix signature is precisely σ−1(K); its surface and basis independence also follows from [F4]. This removes any need to use a root-average convention at −1.

2.1F2F3F4step 1.1algebra

For two successive knots in a crossing-change sequence, use the two corresponding bases from [F2]. If their Seifert matrix difference is ±vvT, their H−1 difference is ±4vvT, of rank at most one; if the difference is zero, the same bound holds. By [F3], the absolute signature change is at most two. Step 1.1 identifies the signatures of these particular matrices with the knot invariants, regardless of which diagram surfaces occur at other stages.

3.1F1F4step 1.1step 2.1choosealgebra∎

By [F1], take a sequence of length u(K) ending at the standard unknot. Its disk Seifert surface has an empty matrix and signature zero. Apply the estimate of step 2.1 to each consecutive pair and add the finitely many inequalities. The triangle inequality yields ∣σ−1(K)∣≤2u(K), including a length-zero sequence. Together with step 1.1 this proves both claims.

Depends on

Used by

Dependency tree · two levels

49 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources