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The figure-eight complement and its nonhyperbolic zero filling

Example

Assume the Axiom of Choice. Let $K_{8}$ be the standard figure-eight knot, let $E_{K_{8}}$ be the exterior obtained by deleting the binding neighborhood from its open-book presentation, and use the positive meridian $μ$ and preferred longitude $λ$ . The interior of the exterior admits a complete finite-volume hyperbolic metric obtained by gluing two regular ideal tetrahedra, of total volume $2 v_{3} = 6 Λ (π / 3) \approx 2.02988$ , where $v_{3} = 3 Λ (π / 3)$ is the volume of one regular ideal tetrahedron and $Λ$ is the Lobachevsky function. The ordinary zero filling along $λ$ is a torus bundle over $S^{1}$ with monodromy matrix $(\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array})$ on the capped fiber homology, and the filled manifold admits no complete hyperbolic metric.

Example references

Danny Calegari, Foliations and the Geometry of 3-Manifolds, Construction 5.37, Theorem 5.38 and Example 5.39

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4, §§4.1–4.4

Diagram

After capping, the punctured-torus fiber becomes $T^{2}$ . The square presents it by identifying equally labelled opposite edges in the indicated directions; $a$ is horizontal and $b$ vertical, so $a \cdot b = + 1$ for the usual square orientation. The curved arrow labels the monodromy action on homology, and the separate circle marks the $S^{1}$ base. The proof is independent of this presentation.

diagram failed to render:
\begin{tikzpicture}[>=stealth,scale=1]
  \draw[->,blue,thick] (0,0) -- node[below] {$a$} (1.5,0);
  \draw[->,blue,thick] (0,1.5) -- node[above] {$a$} (1.5,1.5);
  \draw[->,red,thick] (0,0) -- node[left] {$b$} (0,1.5);
  \draw[->,red,thick] (1.5,0) -- node[right] {$b$} (1.5,1.5);
  \node at (0.75,0.75) {$T^2$ fiber};
  \draw[->,thick] (1.2,1.8) .. controls (1.9,2.9) and (-0.4,2.9) .. (0.3,1.8);
  \node[above] at (0.75,2.8) {$\varphi_* = \begin{pmatrix}2&1\\1&1\end{pmatrix}$};
  \draw[thick] (3.7,0.75) circle (0.30);
  \node[below] at (3.7,0.25) {$S^1$ base};
\end{tikzpicture}

Facts & Assumptions

Given: AC and the standard figure-eight knot with chosen compact exterior and peripheral basis $(μ, λ)$ , with $λ$ the preferred longitude.

[A1]

The Axiom of Choice: AC selects from every family of nonempty sets.

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ selects from every countable family of nonempty sets.

[F2]

AC implies DC implies countable choice: in ZF, $A C \Rightarrow D C \Rightarrow {A C}_{ω}$ .

[F3]

The standard two-tetrahedron figure-eight gluing is complete: the standard two-tetrahedron gluing gives the figure-eight exterior a complete finite-volume hyperbolic metric, with total volume $2 v_{3}$ .

[F4]

Zero filling of a fibered knot caps its fiber: under AC, filling the figure-eight exterior along $λ$ caps its once-punctured torus fiber and gives the torus bundle whose homology matrix is $(\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array})$ .

[F5]

Closed hyperbolic three-manifold groups contain no Z-squared subgroup: under ${A C}_{ω}$ , a closed connected orientable three-manifold with a complete curvature $- 1$ metric has fundamental group with no subgroup isomorphic to $Z^{2}$ .

[F6]

Slopes and generalized fillings on torus cusps: ordinary filling along $λ$ attaches a solid torus with meridian disk boundary in that slope.

[F7]

Covering maps, evenly covered neighbourhoods, fibres, sheets, and trivial coverings: a covering map is a continuous surjection locally given by disjoint sheets, each mapped homeomorphically onto the same open set downstairs.

[F8]

A covering map induces an injective homomorphism on fundamental groups: every covering map induces an injective homomorphism on fundamental groups.

[F9]

The figure-eight complement is a punctured-torus bundle: the standard figure-eight knot is presented as the boundary of the square plumbing of one positive and one negative Hopf band, and deleting its binding neighbourhood gives the stated once-punctured-torus open-book exterior.

Proof

1.1givenA1F1F2F3F4F6F9construct

First identify the two figure-eight models used here. In [F3], $K_{8}$ is the closure of $(σ_{1} σ_{2}^{- 1})^{2}$ , 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.

2.1F7F8step 1.1construct

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

3.1F3F5F6step 1.1step 2.1contradiction∎

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 $Z^{2}$ subgroup, contradicting step 2.1. Hence the zero filling is nonhyperbolic. The original complete structure has volume $2 v_{3} = 6 Λ (π / 3)$ by step 1.1.

Sources

Danny Calegari, Foliations and the Geometry of 3-Manifolds, Construction 5.37, Theorem 5.38 and Example 5.39

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4, §§4.1–4.4

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Sources

Danny Calegari, Foliations and the Geometry of 3-Manifolds (standard reference, not scraped)
William P. Thurston, The Geometry and Topology of Three-Manifolds (standard reference, not scraped)