TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Newton polygon edge slopes are boundary slopes

Statement

Assume full AC and let $K \subset S^{3}$ be a nontrivial smooth knot. Form the Newton polygon of the ordinary A-polynomial using the exponent convention that the lattice point $(a, b)$ represents the monomial $M^{b} L^{a}$ , so edge slopes are measured in the preferred meridian-longitude convention. Every finite rational slope or infinite slope of an edge of this Newton polygon is a boundary slope of an incompressible, boundary-incompressible, orientable surface in the knot exterior. This statement does not assert that the realizing surface is non-fibered or that the slope is strict.

Statement references

Cooper, Culler, Gillet, Long, and Shalen, Theorem 3.4

Tillmann, §7

Facts & Assumptions

Given: Full AC, the nontrivial smooth knot with its chosen compact exterior and peripheral marking, and the stated Newton edge.

[A1]

Full AC is retained. The Axiom of Choice

[F1]

The exterior is compact orientable and irreducible, with no proper compressing disk. Full AC supplies the countable choice needed by its smooth Schoenflies branch. Its tube is a smooth $S^{1} \times D^{2}$ product by the normal-circle frame supplier, so the boundary is the marked smooth torus. It is connected: join two distinct exterior points by an embedded spherical arc, perturb it relative to its endpoints to be transverse to the knot, and use $1 + 1 < 3$ to conclude it misses the knot. Radial projection of nonzero tube fibers to their boundary fixes the exterior and retracts that arc to an exterior path; equal endpoints need the constant path. A smooth knot exterior is irreducible The boundary of a nontrivial knot exterior is incompressible The oriented normal bundle of a knot is trivial Choice-free relative transversality for a compact smooth embedding

[F2]

Every selected Newton edge has an ACTUAL finite representation field and extending local curve valuation with peripheral eigenvalue pair a positive multiple of its primitive minimum normal. For an enhanced edge the representation is absolutely irreducible, including when its factor is also $L - 1$ . Newton edges have valued finite peripheral representation models

[F3]

For that valued representation, the actual lattice-tree/equivariant-map construction gives a connected orientable smooth essential surface with nonempty boundary whose primitive class annihilates the valuation pair. It proves both incompressibility and boundary-incompressibility by map-preserving compression complexity. With an absolutely irreducible representation it also proves that the chosen component cannot be a fiber. Nonelliptic lattice-tree actions produce essential boundary surfaces

[F4]

In the stated marking $a [λ] + b [μ]$ has slope $b / a$ , including $a = 0$ as infinity. Boundary slope of an incompressible surface in a knot exterior

Proof

1.1A1F2givenconstruct

Let $e$ be any edge of the ordinary Newton polygon. Apply [F2] to the ordinary polynomial: it selects an irreducible peripheral curve factor with parallel minimum face, constructs the boundary-branch DVR by the unimodular toric chart, selects an actual generic representation over a finite extension of the peripheral function field by a closed point of the nonzero incidence fiber, and extends the valuation to that field. If $(u, v)$ is the primitive minimum normal in exponent coordinates $(a, b)$ , its eigenvalues satisfy $(w (L), w (M)) = d (u, v)$ for an integer $d > 0$ . Nothing in this step assumes that this ordinary representation is irreducible; it includes the abelian $L - 1$ factor.

2.1F1F3step 1.1construct

The manifold hypotheses of [F3] are exactly [F1]. Its valued-field hypotheses are supplied by step 1.1, and the valuation pair is nonzero. Thus [F3] gives a connected orientable neatly properly embedded smooth surface with nonempty boundary, no compression or boundary compression, and primitive boundary class $(a, b)$ satisfying $a u + b v = 0$ . This is an actual associated midpoint surface after the finite compression and deletion homotopies, not an assumed Culler–Shalen witness.

3.1F2F4step 1.1step 2.1algebra∎

Every direction $(Δ a, Δ b)$ of the original edge also satisfies $u Δ a + v Δ b = 0$ . Consequently its extended rational slope $Δ b / Δ a$ is exactly $b / a$ by [F4], including the vertical case where both denominators vanish and the value is infinity. This is a boundary slope realized by the surface of step 2.1, as required. The proof uses only the ordinary branch of [F2]–[F3] and makes no nonfiber or strictness assertion for that branch.

Depends on

Used by

Enhanced A-polynomial Newton edges give strict boundary slopes Theorem
The two-boundary-slope alternative Theorem

Dependency tree · two levels

26 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

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)
Stephan Tillmann, Varieties associated to 3-manifolds, §7 (standard reference, not scraped)