LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The knot exterior has a peripheral torus basis

Statement

Assume countable choice. For a smooth oriented knot with a chosen product tubular neighbourhood $N (K) ≅ S^{1} \times D^{2}$ , let $μ$ be the positive fibre meridian and let $σ$ be the product section $s \mapsto (s, 1)$ of $T_{K} = \partial N (K) ≅ S^{1} \times S^{1}$ . Then $T_{K} = \partial E_{K}$ is a torus and $[μ], [σ]$ form a basis of $H_{1} (T_{K}; Z)$ . The curve $σ$ is a product section; this statement does not assert that it is the preferred zero-homology longitude.

Statement references

Cattabriga and Francaviglia, §4.4

Hatcher, Example 2.36

Facts & Assumptions

Given: Countable choice, a smooth oriented knot $K$ , its chosen tubular exterior, and the product trivialization of the normal disk bundle.

[A1]

Countable choice, written ${A C}_{ω}$ , is The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F1]

Knot exterior retains $T_{K} = \partial E_{K} = \partial N (K)$ and the tube coordinates as part of the exterior data.

[F2]

Meridian and preferred longitude orients the fibre meridian as the boundary of its oriented normal disk.

[F3]

Tubular exterior and boundary torus supplies $N (K) ≅ S^{1} \times D^{2}$ and $T_{K} ≅ S^{1} \times S^{1}$ under [A1].

[F4]

Cellular homology computes singular homology identifies cellular and singular homology for the finite CW structure on the torus.

[F5]

Cellular homology defines cellular homology as $\ker d_{n} / im d_{n + 1}$ .

[F6]

Cellular boundary from three consecutive skeleta defines the cellular differential by the connecting map of successive skeleta followed by the quotient map.

[F7]

Cellular boundary is the incidence degree matrix computes each cellular boundary coefficient as the degree of the corresponding attaching map after collapsing the other cells.

Proof

technique · direct

1.1A1F1F2F3construct

In product coordinates on $T_{K} = S^{1} \times S^{1}$ , the meridian $μ$ is the second factor loop and the section $σ$ is the first factor loop. Give the torus its standard product CW structure: take the quotient of a square by identifying opposite edges, with one vertex, the two oriented factor loops as its $1$ -cells, and the open square as its $2$ -cell attached by the commutator $μ σ μ^{- 1} σ^{- 1}$ .

2.1F5F6F7step 1.1algebra

The cellular chain groups in dimensions $2, 1, 0$ are $Z, Z^{2}, Z$ . The differential $d_{1}$ is zero because each $1$ -cell starts and ends at the only vertex. By [F7], each coefficient of $d_{2}$ is the degree of the attaching map after collapsing the other $1$ -cell. The commutator traverses each factor once in each orientation, so both degrees are zero. Thus by [F5] and [F6], $H_{1}^{c e l l} (T_{K}; Z) = Z^{2}$ with the factor loops as basis.

3.1F4step 2.1∎

By [F4], the same two classes form a basis of singular $H_{1} (T_{K}; Z)$ .

Peripheral torus diagram

Use coordinates $(u, v) \in R^{2} / Z^{2}$ on the product torus, with $u$ in the oriented knot direction and $v$ in the oriented meridian direction. The square's opposite edges are identified in the same direction. The horizontal and vertical loops are the product section $σ$ and meridian $μ$ ; the diagonal is a representative of $[σ] + [μ]$ . The diagram records the basis convention and is not used in the proof.

Depends on

Used by

An arbitrary parallel push-off need not be preferred Counterexample
Finite-volume hyperbolic knot exterior Definition
Worked case: the unknot exterior Example
Nontrivial knot exteriors are irreducible and boundary-incompressible Lemma
The standard two-tetrahedron figure-eight gluing is complete Lemma
Homology and preferred longitude of a knot exterior Theorem

Dependency tree · two levels

25 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 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology (standard reference, not scraped)