LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 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.

Explicit disk-band knots with arbitrarily small signature bumps

Statement

For every $0 < θ_{0} < π$ and open interval $I \subset (0, π)$ containing $θ_{0}$ , there is a smooth oriented knot $K$ with a supplied compact connected oriented genus-four Seifert surface, ordered integral homology basis and positive-push Seifert matrix $V$ such that $sign ((1 - e^{i θ_{0}}) V + (1 - e^{- i θ_{0}}) V^{T}) = 2,$ and the same signature is zero for all $θ \in [0, π] ∖ I$ . It is nonnegative on $[0, π]$ , constant equal to two on a neighborhood of $θ_{0}$ , and symmetric under $θ \mapsto - θ$ . The surface, basis and knot are given by finite disk-band data; no choice principle or algebraic concordance classification is assumed.

Facts & Assumptions

Given: The parameter $θ_{0}$ and the containing open interval $I$ .

[F1]

An explicit integral eight-dimensional matrix $V = V_{-} \oplus (- V_{+})$ has exactly the claimed Hermitian signature bump, with $V_{\pm} = (\begin{array}{cc} A & I_{2} \\ 0 & B_{\pm} \end{array})$ and symmetric two-dimensional $A, B_{\pm}$ . Each $V_{\pm} - V_{\pm}^{T}$ is $(\begin{array}{cc} 0 & I_{2} \\ - I_{2} & 0 \end{array})$ . Explicit integral Seifert matrices with arbitrarily small signature bumps

[F2]

Every integral matrix with fixed standard two-dimensional symplectic skew blocks has a finite smooth disk-band realization with its exact positive-push Seifert matrix and a specified homology basis, without choice. Finite standard-skew Seifert matrices have explicit disk-band realizations

Proof

1.1F1F2givenconstructalgebra

Take the exact matrix $V$ of [F1]. In the first four-dimensional block, replace the coordinate order $(e_{1}, e_{2}, f_{1}, f_{2})$ by $(e_{1}, f_{1}, e_{2}, f_{2})$ . In the second block, whose skew part is the negative of the first, use $(e_{1}, - f_{1}, e_{2}, - f_{2})$ . These define an explicit signed permutation $P$ with $P^{T} (V - V^{T}) P = ⨁_{j = 1}^{4} (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array})$ . Thus $W = P^{T} V P$ meets [F2] exactly. Apply its finite construction with $g = 4$ to obtain a smooth one-boundary surface and ordered basis with positive-push matrix $W$ . Reorder and reverse the basis cores by the signed permutation $P^{- 1}$ ; their linking numbers change by the corresponding row and column permutations and signs, giving matrix $(P^{- 1})^{T} W P^{- 1} = V$ on the same surface. Its oriented boundary is the required knot $K$ .

2.1F1F2step 1.1algebra∎

This actual surface matrix is exactly the integer matrix in [F1], not just an algebraic concordance class or a rational equivalence class. Its Hermitian matrices therefore coincide entry for entry with those in [F1]. The full signature conclusions, including the values at the two jump parameters and at zero, follow from that finite calculation. The surface has genus four and one boundary component by [F2], and every basis change in step 1.1 was an explicit signed permutation. Only finite constructions and the individual integer witnesses of [F1] were used. This proves the stated knot and supplied-surface realization without invoking an arbitrary Seifert-surface choice or any additional axiom.

Remarks

The conclusion gives an actual knot together with an actual Seifert surface and its signature function. Surface independence, concordance invariance, signature-to-rho formulas, cable reparametrization and companion coefficient-system bookkeeping are separate assertions; none is inferred from this realization lemma.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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

Cha, Structure of the string link concordance group and Hirzebruch-type invariants (standard reference, not scraped)
Cha and Livingston, Knot signature functions are independent (standard reference, not scraped)