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Haken-search acceptance supplies an essential normal disk
Statement
Whenever the Haken branch of the Haken–Jaco–Tollefson search returns a disk-certificate, the associated normal surface is a properly embedded disk in the triangulated knot exterior whose boundary is non-null-homotopic in the exterior's boundary torus.
Statement references
Haken–Jaco–Tollefson normal-surface unknot search — Haken candidate acceptance tests.
Finite normal-coordinate tests characterize essential disk candidates — exact disk and essential-boundary characterization.
Facts & Assumptions
Given: A valid one-component knot code and a vector returned by the Haken branch as a disk-certificate.
Haken–Jaco–Tollefson normal-surface unknot search: the Haken branch returns a vector only after checking matching, quadrilateral admissibility, fundamentalness, nonempty boundary, Euler characteristic , and odd mod-two meridian intersection; fundamentalness gives connectedness by [F2].
Fundamental normal surfaces are connected: a fundamental normal surface is connected.
Finite normal-coordinate tests characterize essential disk candidates: for an admissible normal vector with connected normal surface, the nonempty-boundary, Euler-characteristic and meridian tests pass exactly when that surface is a properly embedded disk whose boundary is non-null-homotopic in the exterior's boundary torus.
Proof
By [F1], the returned vector satisfies the candidate tests in [F1], including normal matching and quadrilateral admissibility.
By [F2], the returned fundamental normal surface is connected. The remaining tests in [F1], together with [F3], identify it as a properly embedded disk in the exterior whose boundary is non-null-homotopic in the exterior's boundary torus.
Depends on
Used by
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Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
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)