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Jaco–Tollefson-search acceptance supplies an essential normal disk
Statement
Whenever the JT branch of the Haken–Jaco–Tollefson search returns a disk-certificate, its primitive vertex normal surface is a properly embedded disk with non-null-homotopic boundary in the triangulated knot exterior.
Statement references
Haken–Jaco–Tollefson normal-surface unknot search — JT primitive-ray enumeration and candidate 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 primitive ray vector returned by the JT branch as a disk-certificate.
Haken–Jaco–Tollefson normal-surface unknot search: a returned JT vector is primitive on an extreme ray, admissible, and passes the nonempty-boundary, Euler-characteristic-one and odd-meridian-intersection tests.
Fundamental normal surfaces are connected: a fundamental normal surface is connected.
Finite normal-coordinate tests characterize essential disk candidates: a connected admissible normal surface passing the nonempty-boundary, Euler-characteristic-one and odd-meridian tests is a properly embedded disk with essential boundary.
Proof
By [F1], the returned vector is admissible and is the primitive integer generator of an extreme ray. If it split into two nonzero admissible integer vectors, both summands would lie on that same extreme ray; primitivity forbids such a positive integral split. Hence the vector is fundamental, and [F2] makes its surface connected.
The remaining tests checked by [F1], together with [F3], identify the connected surface as a properly embedded disk with non-null-homotopic boundary.
Depends on
Used by
Nothing in the library uses this result yet.
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)