Hyperbolic Dehn filling and exceptional slopes — Examples and Counterexamples

The coordinates $(2,0)$ have greatest common divisor $2$, so they are not primitive. By [F1], they therefore do not represent an ordinary slope or the meridian class of an ordinary smooth filling. The primitive pair on the same ray is $(1,0)$.

Write $(2,0)=2(1,0)$, so the radial factor is $r=2$. If a hyperbolic deformation realizes this generalized parameter, [F1] gives its completion cone angle $2\pi \mathrm{/}2=\pi$, rather than the full angle $2\pi$ of an ordinary smooth filling. No realization is asserted or needed: step 1.1 already supplies a nonzero integral pair that cannot be the primitive meridian class of an ordinary filling, refuting the stated universal claim.

First identify the two figure-eight models used here. In [F3], $K_8$ is the closure of $({\sigma }_1{\sigma }_2^{-1}{)}^2$, the standard four-crossing figure-eight diagram. In [F9], the same standard figure-eight knot is presented as the boundary of the square plumbing of a positive and a negative Hopf band; Calegari's Example 5.39 identifies the boundary of the opposite-handed Hopf-band Murasugi sum as the figure-eight knot. Thus both are the same oriented $4_1$ knot class, and an orientation-preserving ambient isotopy identifies the braid closure with the plumbing boundary. Choose the binding and tubular neighborhoods compatibly under this isotopy. It identifies the compact exteriors and carries the positive meridian and preferred longitude to the corresponding peripheral classes (up to simultaneous reversal, which does not change the unoriented zero slope). The hyperbolic exterior from [F3] and the fibered exterior from [F9] are therefore the same marked exterior for this example. By [A1] and [F2], the countable-choice premise [F1] holds. The face gluing and regular-shape assignment in [F3] give the complete finite-volume metric and its two-tetrahedron volume. By [F4] and [F6], the zero filling along the preferred longitude is the stated torus bundle.

Write the bundle as the mapping torus of its torus monodromy. Pull it back along the explicit universal cover $t\mapsto e^{2\pi it}$ of $S^1$; its total space is $T^2\times \mathbb{R}$, and the pullback projection is a covering because it is trivial over each lifted bundle chart. Its fundamental group is ${\pi }_1(T^2)\cong {\mathbb{Z}}^2$. By [F8], this subgroup injects into the fundamental group of the filled manifold.

If the filled manifold admitted a complete hyperbolic metric, it would be a closed connected orientable three-manifold with curvature $-1$: the compact connected exterior has its sole boundary filled by the oriented solid torus of [F6], and the orientation extends across the orientation-reversing boundary attachment. The countable-choice hypothesis holds by step 1.1. By [F5] its fundamental group could contain no ${\mathbb{Z}}^2$ subgroup, contradicting step 2.1. Hence the zero filling is nonhyperbolic. The original complete structure has volume $2v_3=6\mathrm{\Lambda }(\pi \mathrm{/}3)$ by step 1.1.