Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Fundamental normal surfaces are connected
Statement
Every nonempty fundamental normal surface in a finite tetrahedral triangulation of a compact piecewise-linear -manifold is connected.
Statement references
Normal-surface coordinates and the Haken cone — a normal surface has a finite normal-coordinate vector and fundamentalness means no nonzero admissible decomposition.
Hass–Lagarias–Pippenger, Lemma 5.3 — fundamental normal surfaces are connected.
Erickson, “Normal Surfaces and Knots,” §14.3 — connectedness of fundamental normal surfaces.
Facts & Assumptions
Given: A nonempty fundamental normal surface in a finite tetrahedral triangulation.
Normal-surface coordinates and the Haken cone: is an admissible integer vector and fundamentalness excludes every decomposition into two nonzero admissible integer vectors.
Proof
Suppose is disconnected. Since is a finite normal surface, its connected components are finite in number; let be one component and let be the union of all remaining components. Both are nonempty properly embedded normal surfaces: each normal disk and each face-matched arc belongs to exactly one component. Disk counts add coordinatewise, so , and both summands are admissible nonzero integer vectors.
The decomposition in step 1.1 contradicts the defining indecomposability of the fundamental vector . Therefore is connected.
Depends on
Used by
- Haken–Jaco–Tollefson normal-surface unknot search Algorithm
- Finite normal-coordinate tests characterize essential disk candidates Lemma
- Haken-search acceptance supplies an essential normal disk Lemma
- Jaco–Tollefson-search acceptance supplies an essential normal disk Lemma
- Polynomial verification of a primitive normal vertex-disk certificate Lemma
- Correctness of the Haken finite-box disk search Theorem
Dependency tree · one level
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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)