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

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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.

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Finite normal-coordinate tests characterize essential disk candidates

Statement

Let $E$ be a triangulated compact exterior of a smooth knot with peripheral torus $T$ and a marked simple fibre meridian $μ$ in the boundary triangulation. For an admissible normal-coordinate vector $v$ and its associated normal surface $S$ , the following tests hold exactly when $S$ is a connected properly embedded disk with non-null-homotopic boundary in $T$ : (i) $S$ is connected; (ii) $\partial S \neq \emptyset$ ; (iii) $χ (S) = 1$ ; and (iv) $\partial S$ has odd mod-two intersection with $μ$ . If $v$ is fundamental, connectedness follows from fundamentalness.

Statement references

Normal-surface coordinates and the Haken cone — normal-coordinate and admissibility conventions.

Matching equations and finite bounds for normal coordinates — admissible vectors represent normal surfaces.

Fundamental normal surfaces are connected — fundamental normal surfaces are connected.

A connected triangulated surface with Euler characteristic one and boundary is a disk — a connected surface with boundary and Euler characteristic one is a disk.

An essential disk in a knot exterior has longitude boundary — essential disk boundaries in this exterior meet the original meridian oddly.

Facts & Assumptions

Given: The triangulated knot exterior $E$ , its peripheral torus $T$ , marked fibre meridian $μ$ , and admissible vector $v$ .

[F1]

Normal-surface coordinates and the Haken cone: an admissible vector is a nonnegative integer vector satisfying all normal matching equations and the quadrilateral condition.

[F2]

Matching equations and finite bounds for normal coordinates: every admissible integer vector is realized by a properly embedded normal surface with that vector.

[F3]

A connected triangulated surface with Euler characteristic one and boundary is a disk: a connected compact triangulated surface with nonempty boundary and Euler characteristic $1$ is a disk.

[F4]

An essential disk in a knot exterior has longitude boundary: every properly embedded disk with non-null-homotopic boundary in $T$ has boundary $\pm [λ]$ and odd mod-two intersection with $μ$ .

[F5]

Fundamental normal surfaces are connected: every nonempty fundamental normal surface is connected.

Proof

1.1givenF1F2F3construct

The hypothesis that $v$ is admissible has the meaning fixed in [F1]. By [F2], it determines a properly embedded normal surface $S$ , so its boundary lies in $T$ . If tests (i)–(iv) pass, then $S$ is connected, has nonempty boundary, and has Euler characteristic one. By [F3], it is a disk.

1.2givenF4algebra

Conversely, suppose $S$ is a properly embedded disk with non-null-homotopic boundary. A disk is connected, has nonempty boundary, and has Euler characteristic one. By [F4], its boundary has odd mod-two intersection with $μ$ , so all four tests pass.

2.1givenstep 1.1constructcontradiction

If $\partial S$ were null-homotopic in $T$ , its product projection would have degree zero. After a PL transverse perturbation, the mod-two intersection with the fibre meridian equals the parity of the preimages of a regular point under this projection; a lift to $R$ with zero net change crosses that regular level an even number of times. This contradicts test (iv). Thus the disk has non-null-homotopic boundary.

3.1givenF5step 1.1step 1.2∎

If $v$ is fundamental, [F5] already supplies test (i); the other tests remain exactly those stated. Steps 1.1–1.2 prove the equivalence.

Depends on

Used by

Dependency tree · two levels

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Sources

Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger, The Computational Complexity of Knot and Link Problems, Journal of the ACM 46 (1999) (standard reference, not scraped)
Jeff Erickson, Normal Surfaces and Knots, Computational Topology lecture notes (2009) (standard reference, not scraped)