ExampleConstruction: AI-adaptedVerification: 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.

Worked cases of Dehn surgery on the unknot

Example

Assume ${A C}_{ω}$ . Let $U$ be the standard oriented unknot in the oriented $3$ -sphere, with chosen exterior $E_{U}$ and peripheral basis $(μ_{U}, λ_{U})$ . In the orientation convention $L (p, q) = E_{U} (p / q)$ , for coprime integers $p > 0$ and $q$ , the $p / q$ filling is the oriented lens space $L (p, q)$ and has $H_{1} ≅ Z / p Z$ . The $0 / 1$ filling is $S^{1} \times S^{2}$ , the $1 / 1$ filling is $S^{3}$ , and for every integer $n$ the $n / 1$ filling is the boundary of the integer-framed trace $X_{n} (U)$ .

Example references

Countable-choice hypothesis for the chosen peripheral data

Library orientation convention for $L (p, q)$

First-homology formula for knot fillings

Boundary of the integer-framed trace

Saveliev, Lecture 1 §1.5 and Lecture 2 §§2.3–2.4, lens spaces and surgery traces

Ozsváth–Stipsicz–Szabó, §1.2.3 and Exercise 1.2.14, surgery on the unknot

Facts & Assumptions

Given: The unknot exterior $E_{U}$ , its meridian–preferred-longitude basis $(μ_{U}, λ_{U})$ , and the filling convention $p [μ_{U}] + q [λ_{U}]$ .

[F1]

First homology of knot fillings: $H_{1} (E_{K} (p / q); Z) ≅ Z / ∣ p ∣ Z$ for $p \neq 0$ , and $H_{1} (E_{K} (0 / 1); Z) ≅ Z$ .

[F2]

Boundary of the integer-framed knot surgery trace: the outgoing boundary of the integer-framed trace $X_{n} (K)$ is the filling $E_{K} (n / 1)$ .

[F3]

Lens spaces in the $p / q$ filling orientation convention: $L (p, q)$ is the oriented union of the unknot exterior and a solid torus whose meridian maps to $p [μ_{U}] + q [λ_{U}]$ , for $p > 0$ and $\gcd (p, q) = 1$ .

[F4]

Oriented slopes and knot Dehn filling: gluing maps with the same meridian slope give the same oriented filling type.

Verification

1.1givenF3construct

A standard unknot is the coordinate circle in $S^{3} \subset C^{2}$ given by $w = 0$ . Its tubular neighborhood and its exterior are both solid tori; on the exterior, $λ_{U}$ bounds a meridional disk and $μ_{U}$ is a longitude. Therefore each $p / q$ filling is a union of two solid tori, and for $p > 0$ it is $L (p, q)$ by [F3].

2.1givenF4step 1.1construct

When $p = 0$ , primitivity forces $q = \pm 1$ , so the attaching meridian matches the meridian of the exterior solid torus. By the gluing-map independence in the slope definition [F4], use the product gluing; then $S^{1} \times D^{2}$ and $S^{1} \times D^{2}$ glue along $S^{1} \times S^{1}$ as $S^{1} \times (D^{2} \cup_{S^{1}} D^{2}) = S^{1} \times S^{2}$ .

2.2givenF4step 1.1construct

For $p = 1, q = 1$ , the attaching slope is one meridian plus one longitude of the exterior solid torus. A Dehn twist along its meridian disk extends over that torus and changes this slope to its longitude. By [F4] the gluing may then be chosen to be the standard decomposition $S^{3} = (D^{2} \times S^{1}) \cup (S^{1} \times D^{2}) = \partial (D^{2} \times D^{2})$ , so the filled orientation is that of $S^{3}$ .

3.1givenF1step 2.1step 2.2algebra

For all coprime $p, q$ , [F1] gives the stated $H_{1}$ group; the cases $p = 0$ and $p = 1$ agree with steps 2.1 and 2.2.

4.1givenF2step 1.1∎

For every integer $n$ , [F2] identifies the $n / 1$ filling with the outgoing boundary of $X_{n} (U)$ . This adds no rational-framing claim.

Depends on

Used by

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Sources

Nikolai Saveliev, Lectures on the Topology of 3-Manifolds, 2nd revised edition, Lecture 1 §1.5, Lecture 2 §§2.3–2.4 (standard reference, not scraped)
Peter S. Ozsváth, András I. Stipsicz, and Zoltán Szabó, Heegaard Floer homology, version 2025-12-15, §1.2.3 and Exercise 1.2.14 (standard reference, not scraped)