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The two-boundary-slope alternative
Statement
Assume full AC. If is a nontrivial fibered knot that is not a torus knot, then either its exterior contains a connected, separating, incompressible surface whose boundary curves have slope zero, or has at least three distinct boundary slopes.
Statement references
Facts & Assumptions
Given: Full AC and a nontrivial fibered knot in that is not a torus knot.
In the present nontrivial-knot setting, zero is a boundary slope. The ordinary A-polynomial contains ; for the vertical weight its initial form is the product of the other factor’s initial form with this horizontal segment, so its minimum face is a horizontal edge. The ordinary edge theorem therefore supplies an actual incompressible boundary-incompressible orientable realizing surface. Newton polygon edge slopes are boundary slopes
Every enhanced A-polynomial Newton-edge slope has a connected orientable incompressible boundary-incompressible nonfiber realizing surface, by the actual finite valued-model and equivariant compression construction. Enhanced A-polynomial Newton edges give strict boundary slopes
A non-torus knot is not thin by the enhanced-A classification; its enhanced Newton polygon is therefore not contained in a line (The enhanced A-polynomial Newton polygon detects torus knots).
A connected nonseparating incompressible longitudinal surface in this fibered exterior is isotopic to its actual fiber, including the boundary-relative refinement after alignment. The actual proof supplies the proper degree-one surface-map step and the downstairs family-innermost product extraction. A nonseparating incompressible longitudinal surface in a fibered exterior is a fiber
The Newton-edge-to-essential-surface and strictness operations are supplied by [F1]–[F2]. The full enhanced-A classification [F3] now uses its explicit external Baldwin–Sivek Theorem 1.2 input; it does not claim a local reconstruction of gauge theory. Baldwin–Sivek Theorem 1.7 (p.3; §4.1) also states this exact two-boundary-slope alternative, with the same nontrivial fibered nontorus hypotheses.
Proof
By [F3] the enhanced Newton polygon is not contained in a line. A nondegenerate planar polygon has at least two distinct unoriented edge directions, since otherwise all its vertices lie on one line. By [F2] these give two distinct strict boundary slopes.
If there are at least three boundary slopes, the second alternative holds. Otherwise [F1] supplies slope zero, so the two distinct strict slopes from step1.1 exhaust the slope set and one is zero. Choose the connected orientable incompressible nonfiber zero-slope realizing surface supplied by [F2].
If that surface were nonseparating, [F4] would make it isotopic to the fiber, contradicting its nonfiber property. It is separating and supplies the first alternative. These alternatives prove the full Statement, using the explicitly external classification input only at [F3].
Depends on
- The enhanced A-polynomial Newton polygon detects torus knots
- A nonseparating incompressible longitudinal surface in a fibered exterior is a fiber
- Boundary slope of an incompressible surface in a knot exterior
- Enhanced A-polynomial Newton edges give strict boundary slopes
- Newton polygon edge slopes are boundary slopes
Used by
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Dependency tree · two levels
48 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
- John A. Baldwin and Steven Sivek, Torus knots, the A-polynomial, and SL(2,C), Theorem 1.7 and §4.1 (standard reference, not scraped)
- D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen, Plane curves associated to character varieties of 3-manifolds, Theorem 3.4 (standard reference, not scraped)