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Least-weight lexicographic essential normal disks are fundamental
Statement
Let be a compact PL three-manifold with a finite simplicial triangulation . Among its essential normal disks choose one of least edge weight, and among those choose with lexicographically least normal vector. Then is fundamental in the original seven-coordinate system of .
Consequently, if contains a properly embedded PL disk whose boundary is not null-homotopic in , it contains such a disk that is a fundamental normal surface. No ambient orientability, irreducibility or exclusion of projective planes is required, and no choice hypothesis is introduced.
Statement references
Normal-surface coordinates and the Haken cone — admissible normal vectors and fundamentalness in the original triangulation.
Minimum boundary exchange cycles force an impossible zero-edge normal component — the final finite cycle contradiction.
Facts & Assumptions
Given: The finite triangulation and either the extremal normal disk of the first assertion or an essential proper PL disk for the consequence.
Every essential proper PL disk has an essential normal replacement in the same triangulation. (Essential PL disks admit normal disk replacements)
A nontrivial admissible decomposition of a least-weight essential normal disk produces the reduced original-triangulation pair of an inessential-boundary disk and an annulus or Mobius band, with at least two proper double arcs. This reduction includes the finite exclusions of spherical and projective-plane summands. (A nonfundamental least-weight essential disk has a pure boundary obstruction)
Such a reduced boundary pair cannot occur for a least-weight essential disk whose vector is lexicographically least: its finite minimum-leaf cycle produces a nonempty zero-edge component of a genuine partial regular resolution. (Minimum boundary exchange cycles force an impossible zero-edge normal component)
Proof
Suppose the vector of were not fundamental. By Normal-surface coordinates and the Haken cone, it would be for two nonzero admissible integer vectors. Their quad supports are compatible, since their nonnegative sum has at most one quad type in any tetrahedron. Thus [F2] supplies its actual original-triangulation reduced boundary pair. The assumed weight and lexicographic minimality are exactly those of [F3], which excludes that pair. This contradiction proves fundamentalness of .
For the consequence, normalize the given essential disk by [F1]. The set of essential normal disks is therefore nonempty. Its edge weights are positive integers, so choose their least value. Among vectors at this value, lexicographic minimization requires only finitely many natural-number minima: minimize the first coordinate, then the second among those retained, and so on through the coordinates. Each step leaves a nonempty set, and the resulting vector is realized by an essential normal disk. This finite construction uses no selection from an infinite family. Apply step 1.1 to that disk to obtain the claimed fundamental one in the same triangulation.
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Sources
- Wolfgang Haken, Theorie der Normalflächen: Ein Isotopiekriterium für den Kreisknoten, Acta Mathematica105 (1961),245–375 (standard reference, not scraped)