ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

A meridian-preserving longitude twist extends over a solid torus

Example

Let $V = D^{2} \times S^{1}$ have boundary meridian $μ = \partial D^{2} \times {1}$ and longitude $λ = {1} \times S^{1}$ . The boundary matrix $A = (\begin{array}{cc} 1 & 2 \\ 0 & 1 \end{array})$ sends $μ \mapsto μ$ and $λ \mapsto 2 μ + λ$ . Its induced torus homeomorphism extends over $V$ by $(z, w) \mapsto (w^{2} z, w)$ and preserves orientation.

The displayed rectangle is a two-unit-square window in the universal cover $R^{2}$ of the torus $R^{2} / Z^{2}$ , with horizontal coordinate $μ$ and vertical coordinate $λ$ . The identifications are $(x, y) \sim (x + m, y + n)$ for all integers $m, n$ , so each unit square is a fundamental domain. The red lifted segment ends at $(2, 1)$ and therefore represents the image $2 μ + λ$ of the longitude. The picture illustrates the slope convention only; the verification computes the map directly.

Example references

Meridian-preserving extension lemma

Cattabriga and Francaviglia, Knot Theory and Applications: Lecture Notes, §4.1.1

Motegi, Dehn surgery on knots: survey, pp. 19–21

Facts & Assumptions

Given: The product solid torus $V = D^{2} \times S^{1}$ and its meridian-longitude basis $(μ, λ)$ .

[F1]

A torus map carrying the meridian slope to itself extends over the solid torus. A meridian-preserving torus homeomorphism extends over a solid torus

Verification

1.1givenconstructalgebra

The matrix $A$ is integral with determinant $1$ , so its linear map on $R^{2} / Z^{2}$ is a torus homeomorphism. Its first column is $(1, 0)$ , hence it sends $μ$ to $μ$ , and its second column is $(2, 1)$ , hence it sends $λ$ to $2 μ + λ$ .

2.1givenstep 1.1construct

Define $F : V \to V$ by $F (z, w) = (w^{2} z, w)$ . Since $∣ w ∣ = 1$ , this maps $D^{2} \times S^{1}$ to itself, and its inverse is $(z^{'}, w^{'}) \mapsto ((w^{'})^{- 2} z^{'}, w^{'})$ . Thus $F$ is a homeomorphism. On the meridian it has degree $1$ ; on the longitude it winds twice in the disk angle and once in the circle coordinate, giving the same matrix $A$ .

3.1F1step 2.1algebra∎

The disk rotation $z \mapsto w^{2} z$ varies continuously with $w$ and contributes no orientation reversal; the circle coordinate is unchanged. Hence $F$ preserves the product orientation and is the required extension. This also instantiates [F1].

Depends on

A meridian-preserving torus homeomorphism extends over a solid torus

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Alessia Cattabriga and Stefano Francaviglia, Knot Theory and Applications: Lecture Notes, §4.1.1 (standard reference, not scraped)
Kimihiko Motegi, Dehn surgery on knots: survey, pp. 19–21 (standard reference, not scraped)