Thurston hyperbolization for atoroidal knot exteriors

Given: The smooth knot exterior $E_K$ and all hypotheses in the Statement, including full AC. Atoroidality here has the embedded-torus meaning of Incompressibility, Seifert fibrations, and JSJ systems.

External input — Loop Theorem. If a subsurface $F$ of the boundary of a compact $3$-manifold $N$ has noninjective ${\pi }_1(F)\to {\pi }_1(N)$, an essential simple closed curve in $F$ bounds a properly embedded disk in $N$. Source: Aschenbrenner–Friedl–Wilton, 3-Manifold Groups, arXiv:1205.0202, Theorem 1.2, p.7.

External input — comparison of atoroidality conventions. For compact orientable irreducible $3$-manifolds with empty or toroidal boundary, the condition that every incompressible embedded torus is boundary-parallel and the condition that every ${\pi }_1$-injective torus map is homotopic into the boundary agree outside certain Seifert-fibered manifolds. In particular they agree for a manifold which is not Seifert-fibered. Source: the same survey, §1.4, p.9, immediately after its definition of atoroidal. This is the torus-theorem consequence expressly recorded there; it is imported, not reconstructed here.

External input — hyperbolization and the finite-volume case used here. AFW's Hyperbolization Theorem says that a compact, orientable, irreducible $N$ with empty or toroidal boundary, atoroidal in the torus-map convention, and with infinite fundamental group has a complete hyperbolic interior. Source: Theorem 1.13, p.13, with the atoroidality convention on p.9 and the hyperbolic-structure definition on p.11. In the present case, where the torus boundary is incompressible and $N$ is not Seifert-fibered, the finite-volume conclusion is stated explicitly by J. Hyam Rubinstein for compact, orientable, irreducible, atoroidal, non-Seifert manifolds with incompressible torus boundary. Source: “Three-Manifolds with Complete Hyperbolic Metrics of Finite Volume,” §3, p.734. These are imported established results; no general finite-volume conclusion is claimed here for compressible torus boundary.

External input — finite-volume cusp decomposition. A complete oriented hyperbolic $3$-manifold of finite volume is a compact submanifold bounded by tori together with finitely many horoballs modulo rank-two free abelian parabolic actions. Source: Thurston, The Geometry and Topology of Three-Manifolds, Chapter 5, Proposition 5.11.1, printed p.116 (PDF p.34). Thus its noncompact ends are exactly its torus cusps.

The exact cited statements and their indicated conventions, including the finite-volume statement for incompressible torus boundary, were read in the source documents; the short proof of [F4] was also read. The full proofs of the Loop Theorem, the torus theorem and general hyperbolization are external inputs; they have not been locally reconstructed or fully reread for this application.