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TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)
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Jaco–Tollefson normal essential-disk vertex theorem

Statement

Let M be a compact triangulated piecewise-linear 3-manifold with nonempty boundary. If M contains a properly embedded disk whose boundary is not null-homotopic in ∂M, then M contains such a disk whose primitive normal-coordinate vector lies on an extreme ray of the Haken cone.

Statement references

Normal-surface coordinates and the Haken cone — Haken cone, extreme rays and primitive vertex vectors.

Hass–Lagarias–Pippenger, Theorem 5.6 — Jaco–Tollefson vertex-surface theorem.

Facts & Assumptions

Given: The compact triangulated PL three-manifold with nonempty boundary and the stated essential properly embedded disk.

[F1]

An essential proper PL disk has an essential normal replacement in the same triangulation. Essential PL disks admit normal disk replacements

[F2]

A least-weight essential normal disk with lexicographically least vector has primitive vector on an extreme ray of the original full Haken cone. Its proof handles both original two-copy summand alternatives and all proper exchanges without irreducibility, orientability, sphere-separation or projective-plane hypotheses. Least-weight lexicographic essential normal disks are vertex disks

[F3]

External input (PL Loop Theorem). For a triangulated PL three-manifold, not necessarily orientable, a continuous map (D2,∂D2)→(M,∂M) with essential boundary implies existence of a proper PL embedded disk with essential boundary. This is the PL form of Hatcher, Theorem 3.1 and its finite tower/surgery proof, pp.56–59; no relative approximation of a wild embedded disk is asserted.

Proof

1.1givenF1F3construct

Regard the supplied disk inclusion as a continuous map of pairs and apply [F3]. This supplies an essential proper PL disk, without asserting that it is the original disk. Normalize that disk by [F1]. Choose the least positive edge weight among essential normal disks and then minimize the finitely many coordinates successively at that weight. This uses only natural-number minima, not a choice principle.

2.1givenF2step 1.1construct∎

The selected disk satisfies [F2]. It is essential and normal in the same original triangulation, and its normal vector is primitive and lies on an extreme ray of the Haken cone. These are exactly the required conclusions for arbitrary compact PL input.

Depends on

Used by

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Sources