CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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The core pattern has a product companion torus

Statement refuted

For every nontrivial companion knot and every pattern in its solid torus, the companion torus in the satellite exterior is essential.

Statement references

Satellite pattern, companion, and winding number

Facts & Assumptions

Given: The pattern, the companion embedding, and the two nested tubular neighbourhoods specified in the Counterexample.

[F1]

Satellite pattern, companion, and winding number defines the core pattern, satellite knot, pattern exterior, and companion torus.

[F2]

Incompressibility, Seifert fibrations, and JSJ systems defines boundary-parallel torus and essential torus.

Counterexample

Given: A nontrivial knot $K$ with companion solid torus $V = e (S^{1} \times D^{2})$ .

1.1F1givenconstruct

Take the core pattern $P = S^{1} \times {0} \subset V$ and choose its tubular neighbourhood $N (P)$ to be a smaller concentric solid torus. The companion embedding sends this core to $K$ , so $P (K) = K$ with the smaller chosen tube, while the companion torus is the boundary of the larger tube.

2.1F1F2step 1.1construct∎

Radial interpolation in the $D^{2}$ coordinate identifies the shell between the two concentric tubes with $T^{2} \times [0, 1]$ . It lies in the satellite exterior and has the companion torus and the satellite exterior boundary as its ends. Thus the companion torus is boundary-parallel and is not essential by [F2]. This refutes the unrestricted statement.

Remarks

This example shows why a nonproduct condition on the pattern exterior is needed in the essential-companion-torus lemma.

Remarks references

Essential companion tori for nontrivial companions and incompressible nonproduct patterns

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

Allen Hatcher, Notes on Basic 3-Manifold Topology (standard reference, not scraped)