A new family of minimal ideal triangulations
of cusped hyperbolic 3–manifolds

We refer to [10] for background and precise definitions used in this paper and only give a quick summary here. For a cusped orientable hyperbolic 3–manifold $M$ of finite volume, the norm $||\ \alpha\ ||$ of a non-trivial class $\alpha\in H_{2}(M,\mathbb{Z}_{2})$ is the negative of the maximal Euler characteristic of a properly embedded surface $S$ (no component of which is a sphere) representing the class. Any surface $S$ with $[S]=\alpha$ and $-\chi(S)=||\ \alpha\ ||$ is taut if no component of $S$ is a sphere or a torus. A class $\alpha\in H_{2}(M,\mathbb{Z}_{2})$ determines a labelling of the ideal edges of an ideal triangulation of $M$ by elements in $\mathbb{Z}_{2}=\{0,1\}$. These can be interpreted as edge weights of a normal surface, called the canonical normal representative of $\alpha.$

The complexity $c(M)$ is the minimal number of ideal tetrahedra in an ideal triangulation of $M.$ In [10, Theorem 1], the authors and Bus Jaco obtain the following lower bound on the complexity of a cusped hyperbolic $3$-manifold:

It is shown in [10] that the above bound implies that the monodromy ideal triangulations of the hyperbolic once-punctured torus bundles are minimal. This note presents another infinite family of once-cusped hyperbolic $3$-manifolds $M_{k,n}$, admitting triangulations $\mathcal{T}_{k,n}$ which achieve the lower bound in Theorem 1. These manifolds are shown to arise from Dehn surgery on the link $8^{3}_{9}.$ What makes these triangulations interesting is that their structure is different from the structure of the monodromy ideal triangulations, and that they do not feature layered solid tori as Dehn fillings (as, for instance, the canonical triangulations exhibited by Guéritaud and Schleimer [7]). We now state the main theorem of this paper (the relevant definitions and an outline of the proof are given in Section 2):

This theorem, and its proof, have the following consequences.

First, our approach to compute the $\mathbb{Z}_{2}$–norm introduces new techniques that extend the tool kits from [9, 10, 11] and is also applicable to the computation of the Thurston norm of Dehn fillings.

Second, the construction of the minimal triangulations generalises. We do not provide full details, and simply list a further family of triangulations that we conjecture to be minimal in Section 7.

Third, it remains an open problem to classify all ideal triangulations of cusped hyperbolic 3–manifolds that achieve the lower bound in Theorem 1, and to provide a complete list of all underlying cusped hyperbolic 3–manifolds. In contrast, an analogous bound for the complexity of closed 3–manifolds and a complete characterisation of all tight examples are given in [9].

Acknowledgements. Research of the authors is supported in part under the Australian Research Council’s Discovery funding scheme (project number DP190102259).

The first member, $\mathcal{T}_{3,3}$, of the infinite family $\mathcal{T}_{k,n}$ was found by experimentation. Of the $9\,596$ minimal triangulations of $4\,815$ cusped hyperbolic $3$–manifolds contained in the orientable cusped hyperbolic census (shipped, for instance, with Regina [3]), there are exactly nine manifolds for which the lower bound in Theorem 1 is attained. Eight of these are monodromy ideal triangulations of once-punctured torus bundles, but the remaining one, a triangulation of manifold s781, is not.

One can check that this minimal ideal triangulation $\mathcal{T}_{3,3}$ of s781 achieves the lower bound as follows. It has isomorphism signature gLLMQbeefffehhqxhqq and six ideal tetrahedra. The underlying hyperbolic manifold $M_{3,3}$ has one cusp and $H_{1}(M_{3,3},\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{4}.$ The space of all closed normal surfaces has 11 embedded fundamental surfaces consisting of one vertex linking torus, seven orientable surfaces of Euler characteristic at most $-2$ and three non-orientable surfaces of Euler characteristic equal to $-2.$ The non-orientable surfaces represent distinct non-trivial classes in $H_{2}(M_{3,3},\mathbb{Z}_{2}).$ Since each even torsion class has a taut representative that is isotopic to a normal surface, and all non-trivial fundamental normal surfaces in $\mathcal{T}_{3,3}$ have Euler characteristic $\leq-2$, it follows that $\mathcal{T}_{3,3}$ attains the lower bound in Theorem 1.

We now briefly describe how this example is extended to the infinite family of triangulations $\mathcal{T}_{k,n}$, $k,n\geq 3$ odd.

The ideal triangulation $\mathcal{T}_{k,n}$ is build from a $k$–tetrahedron solid torus $T_{k}$ and an $n$–tetrahedron solid torus $T_{n}$, $k,n\geq 3$ odd, each with two punctures on the boundary, by identifying the eight boundary triangles (four per solid torus) in pairs. The identification space $M_{k,n}$ has the required torsion classes in homology, namely $H_{1}(M_{k,n},\mathbb{Z})\cong\mathbb{Z}\oplus\mathbb{Z}_{2}\oplus\mathbb{Z}_{k+1}$ (note that $H_{1}(M_{k,n},\mathbb{Z})$ does not depend on $n$). Hence we have $H_{2}(M_{k,n},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$. We remark that we do not consider homology relative to the boundary. The triangulations of the solid tori are described in Section 3, and the triangulations $\mathcal{T}_{k,n}$ in Section 4.

In order to apply Theorem 1, we need to show that each $M_{k,n}$ is hyperbolic, i.e. has a complete hyperbolic structure of finite volume. To this end, we show in Section 5 that the manifolds $M_{k,n}$ are obtained by Dehn surgery on the link $8^{3}_{9}$ and determine the surgery coefficients. The complement of the three component link $8^{3}_{9}$, shown in Figure 1, is hyperbolic and admits a symmetry which fixes one of the cusps and interchanges the other two. We refer to the former as the symmetric cusp, and it is shown in red in Figure 1. The manifolds $M_{k,n}$ are all obtained from the complement of $8^{3}_{9}$ by filling two of its cusps, where one of these must be the symmetric cusp. An application of Lackenby’s combinatorial word hyperbolic Dehn surgery theorem [14, Theorem 4.9] shows that $M_{k,n}$ is hyperbolic for all $k,n\geq 17.$ A computation with Regina [3] extends this to all $k,n\geq 3.$

We next determine the norm of the non-trivial classes in $H_{2}(M_{k,n},\mathbb{Z}_{2}).$ This is achieved in Section 6 by using a different triangulation of $M_{k,n}$, namely a triangulation $\mathcal{T}^{\prime}_{k,n}$ that uses the description of $M_{k,n}$ as Dehn surgery on the link $8^{3}_{9}.$ This allows us to analyse the space of all normal surfaces representing the homology classes. Namely, it reduces the argument to a calculation of all normal surfaces with a specific boundary pattern in the link complement, together with an analysis of how these surfaces extend to the surgery tori.

This completes the outline of the proof of Theorem 2. Below is a quick guide to the notation used for the various triangulations in this paper:

The construction of our infinite families of minimal triangulations of once-cusped hyperbolic manifolds is based on a family of solid torus triangulations $T_{m}$. Start with a 2–triangle standard triangulation of the annulus. The two boundary curves of the annulus are edges in this triangulation, and each of the remaining three edges is incident with both boundary edges. One now iteratively layers tetrahedra on one side of this annulus, such that the initial triangulation of the annulus is in the boundary of the solid torus, and each interior edge has degree four. This completely characterises these triangulations combinatorially. In what follows, we mainly introduce notation to make this explicit and we highlight some features that will be relevant later.

Let $m\geq 1$ and $\Delta_{1},\dots,\Delta_{m}$ be a collection of Euclidean 3–simplices. The vertices of each $\Delta_{j}$ are labelled $0,1,2,3$, and for each subset $\Lambda\subseteq\{0,1,2,3\},$ $\Delta_{j}(\Lambda)$ is the subsimplex spanned by $\Lambda.$ The triangulation $T_{1}$ is obtained by making the edge identification $\Delta_{1}(03)=\Delta_{1}(12).$ For $m>1,$ $T_{m}$ is obtained from $T_{m-1}$ by adding $\Delta_{m}$ via the face pairings:

If $m$ is even, this is a layering of $\Delta_{m}$ on the edge $\Delta_{m-1}(03)$; and if $m$ is odd, this is a layering of $\Delta_{m}$ on the edge $\Delta_{m-1}(13).$

It follows from the construction that the faces $\Delta_{1}(012)$ and $\Delta_{1}(023)$ form an annulus in the boundary of $T_{m}$, and that for $m>1,$ $T_{m}$ is a solid torus obtained from $T_{m-1}$ by layering a tetrahedron on an edge not contained in the boundary faces $\Delta_{1}(012)$ and $\Delta_{1}(023)$. The different pairings for $m$ odd or even ensure that each interior edge of $T_{m}$ has degree four.

The two boundary edges of $T_{m}$ corresponding to a longitude of the solid torus are $\Delta_{1}(01)=\ldots=\Delta_{m}(01)$ and $\Delta_{1}(23)=\ldots=\Delta_{m}(23)$. We denote $\Delta_{1}(01)=\ldots=\Delta_{m}(01)$ by $\lambda$, see Figure 2.

We note that $T_{m}$ has two vertices $\Delta_{1}(0)=\Delta_{1}(1)=\ldots=\Delta_{m}(0)=\Delta_{m}(1)$ and $\Delta_{1}(2)=\Delta_{1}(3)=\ldots=\Delta_{m}(2)=\Delta_{m}(3)$.

The boundary of the solid torus $T_{m}$ is a 4–triangle, 2–vertex torus. For $m$ odd this results in boundary faces $\Delta_{m}(013),$ $\Delta_{m}(123),$ $\Delta_{1}(012),$ and $\Delta_{1}(023),$ see also Figure 2.

In order to determine the meridian $\mu$ of $T_{m}$ we trace the meridian disc of $T_{m}$ through the layerings:

• •

  In the 2–triangle annulus the seed of the meridian disc is a horizontal arc connecting the two boundary components (Figure 2 top row).

• •

  Layering $\Delta_{1}$ to the annulus adds a single horizontal quad of type $q_{03/12}$ in $\Delta_{1}$ to the seed of the meridian disc (Figure 2, second row).

• •

  Layering $\Delta_{2}$ on edge $\Delta_{1}(03)=\Delta_{1}(12)$ adds two normal triangles in $\Delta_{2}$ (one of type $t_{0}$ and one of type $t_{3}$) to the meridian disc (Figure 2, third row).

• •

  Layering $\Delta_{3}$ on edge $\Delta_{2}(02)=\Delta_{2}(13)$, adds two normal triangles (of types $t_{1}$ and $t_{3}$ respectively) and one normal quadrilateral (of type $q_{01/23}$) in $\Delta_{3}$ to the meridian disc (Figure 2, bottom row).

• •

  Layering $\Delta_{j}$, $j\geq 4$, adds two more normal triangles (of types as before) and $m-2$ normal quadrilaterals of type $q_{01/23}$ to the meridian disc. See Figure 3 for the meridian disc in $T_{m}$ and its boundary curve.

The triangulation $T_{m}$ admits three normal surfaces $F_{1}$, $F_{2}$, and $F_{3}$ with exactly one normal quadrilateral per tetrahedron and every normal quadrilateral type features in exactly one of these three surfaces:

• •

  Starting with quadrilateral $q_{01/23}$ in $\Delta_{1}$ we obtain a separating annulus $F_{1}$ intersecting the boundary of $T_{m}$, $m\geq 2$ arbitrary, in two longitudes, denoted by $\gamma_{1}$.

• •

  Starting with quadrilateral $q_{02/13}$ in $\Delta_{1}$ we obtain a one-sided (non-orientable) surface $F_{2}$ intersecting the boundary of $T_{m}$, $k\geq 3$ and odd, horizontally in curve $\gamma_{2}$.

• •

  Starting with quadrilateral $q_{03/12}$ in $\Delta_{1}$ we obtain another one-sided (non-orientable) surface $F_{3}$ intersecting the boundary of $T_{m}$, $m\geq 3$ and odd, diagonally in curve $\gamma_{3}$.

See Figure 4 for an illustration.

Due to the following result, we know when and how a curve on the boundary of a solid torus extends to a one-sided incompressible properly embedded (connected) surface in its interior.

Suppose $k,n\geq 3$ odd. Let $T^{1}=T_{k}^{1}=T_{k}$ and $T^{2}=T_{n}^{2}=T_{n}$ be triangulations of the solid torus from Section 3. Denote their tetrahedra by $\Delta^{i}_{j},$ $i=1,2.$ Glue the boundary of $T^{1}$ to the boundary of $T^{2}$ in the following way:

We denote the set the face pairings in (4.1) by $\Phi.$

This identifies all vertices from both boundary tori, and the link of this single vertex is a torus. The identification space of the resulting triangulation $\mathcal{T}_{k,n}$ is an orientable pseudo-manifold having the single vertex as the only non-manifold point, and the complement of the vertex is denoted $M_{k,n}.$ See Figure 10 for the two torus boundaries of $T^{1}$ and $T^{2}$.

We refer to the image of the boundaries of $T^{1}$ and $T^{2}$ in $\mathcal{T}_{k,n}$ as the gluing interface, see Figure 6 for a drawing of the associated abstract $2$-complex. As observed, this has four triangles, one vertex, and we claim that there are four edges. For the twenty edges involved in the identifications on the boundaries of the tori, we have identifications $\Delta_{1}^{1}(01)=\Delta_{k}^{1}(01)$, $\Delta_{1}^{1}(23)=\Delta_{k}^{1}(23)$, $\Delta_{1}^{1}(03)=\Delta_{1}^{1}(12)$ and $\Delta_{k}^{1}(03)=\Delta_{k}^{1}(12)$ coming from the gluings inside $T^{1}$; and similarly $\Delta_{1}^{2}(01)=\Delta_{n}^{2}(01)$, $\Delta_{1}^{2}(23)=\Delta_{n}^{2}(23)$, $\Delta_{1}^{2}(03)=\Delta_{1}^{2}(12)$ and $\Delta_{n}^{2}(03)=\Delta_{n}^{2}(12)$ coming from the gluings inside $T^{2}$. Combining this with the identifications of the boundaries yields four edge classes in the gluing interface:

Denote the quadrilateral normal surfaces in the solid tori $T^{i}$ described in Section 3 by $F^{i}_{j}$, $i=1,2$, $j=1,2,3$. Their edge weights $(w(e_{1}),w(e_{2}),w(e_{3}),w(e_{4}))$ in $\partial T^{i}$, $i=1,2$ are

• •

  $(1,1,0,1)$ for $F_{2}^{1}$ and $F_{1}^{2}$;

• •

  $(0,0,1,1)$ for $F_{1}^{1}$ and $F_{3}^{2}$;

• •

  $(1,1,1,0)$ for $F_{3}^{1}$ and $F_{2}^{2}$.

It follows that $\mathcal{T}_{k,n}$ admits three normal surfaces $\mathcal{S}_{1}=F_{2}^{1}\cup F_{1}^{2}$, $\mathcal{S}_{2}=F_{1}^{1}\cup F_{3}^{2}$, and $\mathcal{S}_{3}=F_{3}^{1}\cup F_{2}^{2}$, each consisting of a single quadrilateral per tetrahedron such that every normal quadrilateral type of $\mathcal{T}_{k,n}$ occurs in exactly one of these surfaces.

Their respective Euler characteristics can be computed from Proposition 3. For this, note that the boundary curves of the surfaces $F^{i}_{j}$ inside $T^{i}$, $i=1,2$, $j=1,2,3$, are made up of four normal arcs and four midpoints of edges.

In $\mathcal{T}_{k,n}$, the surfaces $\mathcal{S}_{j}$, $j=1,2,3$, still meet the gluing interface $T^{1}\cap T^{2}$ in four normal arcs, but only three (resp. two) midpoints of edges for $\mathcal{S}_{3}$ and $\mathcal{S}_{1}$ (resp. $\mathcal{S}_{2}$). This can be seen by looking at their edge weights for the four edges in the gluing interface.

Thus, the Euler characteristic of each surface pieces is the value given in Proposition 3 minus $1$ (resp. minus $2$). Altogether we have

Each of the $\mathcal{S}_{j}$, $j=1,2,3$ is the canonical representative of one of the non-zero $\mathbb{Z}_{2}$–homology classes and we have

In order to apply Theorem 1 to conclude that $\mathcal{T}_{k,n}$ is minimal, we need to show that each $M_{k,n}$ is hyperbolic (see Section 5) and that each $\mathcal{S}_{j}$ is taut (see Section 6).

We start by building a 3–cusped manifold using a variant of the construction of $\mathcal{T}_{k,n}$ described above. For this, take a cone $\widehat{C}^{1}$ over $\partial T^{1}$ and a cone $\widehat{C}^{2}$ over $\partial T^{2}$, where $T^{1}=T_{k}^{1}$ and $T^{2}=T_{n}^{2}$ are the solid torus triangulations from Section 4 and $k,n\geq 3$ both odd. Now we glue them together using the same face pairings $\Phi$ from (4.1) to obtain a triangulation $\mathcal{T}$ with eight tetrahedra of a pseudo-manifold $\widehat{N}$ with three vertices having torus links. This is defined by the gluing table given in Table 1 and shown in Figure 5.

Using Regina [3] we can check that the complement $N=\widehat{N}\setminus\widehat{N}^{(0)}$ of the three vertices in $\widehat{N}$ is a 3–manifold homeomorphic to the complement of the link $8^{3}_{9}$, and that this is a minimal triangulation of $N.$ We refer to $C^{1}=\widehat{C}^{1}\setminus\widehat{N}^{(0)}$ and $C^{2}=\widehat{C}^{2}\setminus\widehat{N}^{(0)}$ respectively as the (red) cusp 1 and the (blue) cusp 2 of $N.$ The red cusp is the symmetric cusp of the link complement, and the blue cusp therefore one of the non-symmetric cusps.

It follows from the construction that the interface between the red and blue cusps in $N$ is identical to the interface between the solid tori in $M_{k,n}.$ It is shown in Figure 6. We now truncate the red and blue cusps along vertex linking surfaces. These surfaces are naturally combinatorially isomorphic with $\partial T^{1}$ and $\partial T^{2}$. The cusps are then filled with the solid tori $T^{1}_{k}$ and $T^{2}_{n}$ respectively, such that the gluing agrees with the natural combinatorial isomorphism (see Figure 7). With respect to the framing indicated in the figure, this results in the manifold

For the interested reader who wishes to use SnapPy to study these manifolds, we remark that

The Euclidean equilateral triangles shown in Figure 7 in fact give the Euclidean structures (up to scale) on the cusp cross sections with respect to the complete hyperbolic structure on $N.$ Here, each ideal tetrahedron has shape $i$ and each triangle therefore angles $\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{4}$. The framing used in this paper is thus the natural geometric framing of the cusps.

Casson observed that if an ideal triangulation admits an angle structure, then the manifold must admit a complete hyperbolic metric of finite volume (see [16, Theorem 10.2]). Using this, computation with Regina [3] confirms that $N_{k,n}$ is hyperbolic for all $17\geq k,n\geq 3.$ The same result could also be obtained with SnapPy [4].

To show that $N_{k,n}$ also has a complete hyperbolic metric of finite volume for all $k,n\geq 17,$ we note that due to Thurston’s hyperbolisation theorem for Haken manifolds [18, 15, 13], it suffices to show that $N_{k,n}$ is irreducible, atoroidal, and not Seifert fibred. This conclusion is achieved by an application of Lackenby’s combinatorial version of the Gromov-Thurston $2\pi$ theorem [14, Theorem 4.9] (see also the sentence after its proof for the case when not all cusps are filled). We refer to [14] for the definitions and statements required to understand the next paragraph.

The hypothesis of [14, Theorem 4.9] is achieved by applying [14, Proposition 4.10], which gives a criterion using angle structures. We use the angle structure on $N$ determined by the complete hyperbolic structure, where each triangle in the vertex link has angles $\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{4}$. In particular, Lackenby’s combinatorial length of any curve $\gamma$ on a vertex link is estimated from below by the shortest simplicial length $|\gamma|$ of a representative of $[\gamma]$ times $\frac{\pi}{8}.$ The theorem thus applies to any set of filling curves $\gamma$ with $|\gamma|\frac{\pi}{8}>2\pi,$ and hence $|\gamma|\geq 17.$ Examination of the vertex links gives the simple bounds $|\mu_{1}^{-k}\lambda_{1}|\geq k$ and $|\mu_{2}^{n+1}\lambda_{2}|\geq n+1$. Therefore $k,n\geq 17$ are sufficient to ensure hyperbolicity.

Next, we observe that by construction the manifold $N_{k,n}$ is homotopy equivalent with $M_{k,n},$ and that we may choose the homotopy to be the identity in a neighbourhood of the single cusp (corresponding to the vertex on the interface). Now take a sufficiently large compact core of each manifold, and double it along its boundary. Then we have two closed Haken 3–manifolds that are homotopy equivalent, and hence homeomorphic by Waldhausen [19]. Since $N_{k,n}$ is hyperbolic and homeomorphism preserves the JSJ decomposition, it follows that $N_{k,n}$ is homeomorphic with $M_{k,n}.$ In particular, $M_{k,n}$ is also hyperbolic.

It is clear from the description of the normal surfaces in the previous section that $M_{k,n}=N_{k,n}$ has homology amenable to an application of Theorem 1 ([10, Theorem 1]). This can be verified using a homology computation using the chosen peripheral framings. Using the triangulation, one computes

with the two peripheral subgroups of interest to us (in the geometric framing) generated by:

This implies:

In particular, this highlights the requirement for both $n$ and $k$ to be odd.

In this section we determine the norm of the non-trivial classes in $H_{2}(M_{k,n},\mathbb{Z}_{2})$. We first provide an overview. Our proofs of correctness of the calculation rely on technical results by Jaco and Sedgwick [12] for normal surfaces with boundary on a single 2–triangle torus and Bachman, Derby-Talbot and Sedgwick [1] for normal surfaces with boundary on two 2–triangle tori.

We consider a triangulation $\mathcal{T}^{\prime}$ of the 3–manifold obtained by removing open neighbourhoods of the blue and red cusps of the complement of $8^{3}_{9}$ (Section 6.1). This has one cusp and two boundary components, $\partial_{1}$ (the boundary of the blue cusp) and $\partial_{2}$ (the boundary of the red cusp). The interior of the identification space $\overline{\mathcal{T}}$ of $\mathcal{T}^{\prime}$ is homeomorphic to the complement of $8^{3}_{9}.$ Using the natural geometric framing on $\partial_{1}$ and $\partial_{2}$ and the surgery description of $M_{k,n}$, we determine how to complete $\mathcal{T}^{\prime}$ to a triangulation $\mathcal{T}^{\prime}_{k,n}$ of $M_{k,n}$ by gluing suitable layered solid tori to $\partial_{1}$ and $\partial_{2}$ (Section 6.5).

Each non-trivial class $\alpha\in H_{2}(M_{k,n},\mathbb{Z}_{2})$ associates with each edge in $\partial_{1}$ and $\partial_{2}$ a parity, which we call its boundary pattern (Section 6.2). This boundary pattern encodes the parities of the number of points in the intersection of the edges with a surface that is transverse to the triangulation and represents the class $\alpha.$ There is a taut normal surface $S$ in $\mathcal{T}^{\prime}_{k,n}$ that represents the class $\alpha$ and its intersection with each of $\partial_{1}$ and $\partial_{2}$ consists of at most one curve, and these curves of intersection are essential ( Section 6.3).

The norm of $\alpha$ is the maximum Euler characteristic of any normal surface in $\mathcal{T}^{\prime}_{k,n}$ representing $\alpha.$ The computation of the norm of $\alpha$ thus splits into a computation of all normal surfaces of $\mathcal{T}^{\prime}$ that have the correct boundary pattern (Section 6.3) and determining how these normal surfaces extend into the layered solid tori attached to $\mathcal{T}^{\prime}$ along $\partial_{1}$ and $\partial_{2}$ (Section 6.6). Normal surfaces in these layered solid tori are well-known (Section 6.4). Moreover, the Euler characteristics of the latter surfaces are determined by their boundary slopes via the Bredon-Wood formula (see Proposition 3 or [5, Corollary 2.2]). The conclusion of these computations in Section 6.6 is that:

In particular, this confirms that the surfaces $\mathcal{S}_{1},\mathcal{S}_{2}$ and $\mathcal{S}_{3}$ described in Section 4 are taut. We now supply the missing details.