The Complete 10-Tetrahedra Census of Orientable Cusped Hyperbolic 3-Manifolds

A (generalised) triangulation $𝒯$ is a disjoint union of finitely many tetrahedra, with some or all faces identified in pairs via simplicial maps, and we avoid identifications where an edge is identified with itself in reverse. Note that $𝒯$ is not necessarily a simplicial complex, as identifications of two faces of the same tetrahedron are allowed.

Quotienting by the identifications, $𝒯$ can be seen as a topological space, along with a cellular structure induced by the tetrahedra. A vertex in $𝒯$ is a $0$-cell in this cellular structure, and an edge is a $1$-cell. The link of a vertex in $𝒯$ is the frontier of a small regular neighborhood of that vertex. A vertex is internal if its link is a sphere, and is ideal if its link is some other closed surface. For an edge $e$ in $𝒯$, its degree is the cardinality of the set

where ${𝒩}_e$ is a small regular neighborhood of the midpoint of $e$.

It follows that every vertex of an ideal triangulation is either internal or ideal. Note that an ideal triangulation itself is not a manifold and has at least one ideal vertex.

If $M$ is a cusped hyperbolic $3$-manifold, an ideal triangulation $𝒯$ of $M$ is geometric if all vertices are ideal, all edges are geodesics, all $2$-cells are parts of geodesic planes, and no flat or negatively oriented tetrahedron (i.e. with zero or negative volume) exists.

The following results were proven in Section 3 of [7]:

In a tetrahedron, an arc $\alpha$ properly embedded in one of its faces is normal if the endpoints of $\alpha$ are in the interior of distinct edges. A properly embedded disk is a triangle if its boundary consists of three normal arcs in distinct faces, and is a square if its boundary consists of four normal arcs in distinct faces.

Let $𝒯$ be a triangulation. A properly embedded compact surface in $𝒯$ is normal if its intersection with every tetrahedron in $𝒯$ (with the identifications undone) is a properly embedded surface whose connected components consist of only triangles and squares.

An isotopy of $𝒯$ is normal if it preserves the cellular structure of $𝒯$. Up to normal isotopy, there are $4$ triangles and $3$ squares in a tetrahedron. Multiple copies of triangles or squares in the same normal isotopy class may appear in the intersection of a normal surface and a tetrahedron. It follows that a normal surface in a triangulation of $n$ tetrahedra is uniquely determined by a vector in ${\mathbb{N}}^{7n}$ up to normal isotopy, where $\mathbb{N}≔\{k\in \mathbb{Z}\mid k\ge 0\}$.

The subset $𝒩≔\{𝒗\in {\mathbb{N}}^{7n}\mid 𝒗\text{ represents a normal surface}\}$ consists of integral vectors in a certain non-convex subset of a polyhedral cone in ${ℝ}^{7n}$ [29]. The standard vertex normal surfaces correspond to vertices of a cross-section of this polyhedral cone, and can be viewed as forming a finite basis for $𝒩$.

The following result was proven in Section 3 of [7]:

To build a census (of knots, manifolds, etc), the procedure generally follows a $3$-step outline:

1. (i)

   Generate all candidates for the census.

2. (ii)

   Decide the eligibility of each candidate.

3. (iii)

   Group eligible candidates into isomorphism classes and certify that the isomorphism classes are all distinct.

As long as the candidate generation is performed exhaustively and the eligibility for each candidate is decided correctly, the census obtained is guaranteed to contain all wanted objects. With step (iii), the resulting census will contain no duplicates. We say that a census is complete if it contains all wanted objects without any duplicated ones.

For censuses of cusped hyperbolic $3$-manifolds, the candidates will be ideal triangulations consisting of a given number of tetrahedra, and the eligibility conditions will be minimality (of triangulations) and hyperbolicity (of manifolds the triangulations represent). For certifying distinctness, algebraic invariants were widely applied for both the census of hyperbolic $3$-manifolds [7] and the census of knots [28, 8, 43]. However, it is unknown whether algebraic methods will indeed separate all given isometry classes within a given scope, consequently they cannot be used to certify if two isometry classes are indeed the same, and other methods will be required for that purpose. Here we propose an alternative approach for cusped hyperbolic $3$-manifolds, using a geometric invariant, the canonical triangulation, built upon the Epstein–Penner canonical cellulation [21] by Weeks [44], whose rigorous computation was implemented by Goerner [24]. An introduction to these can be found in [22], and we review them briefly in Section 3.3.1. The canonical triangulation, when computed rigorously, is a complete invariant of cusped hyperbolic $3$-manifolds, and we applied it to deduplicate a majority of triangulations in the $10$-tetrahedra census in Section 3.3.3.

Our approaches for steps (i) and (ii) largely overlap with those in [7]. For completeness, we briefly describe them in Section 3.2, and kindly refer the reader to [7] for more details. For step (iii), detailed discussions will be given in Section 3.3.

Let us start by looking at Table 2. The steps proceed from top to bottom. Each row consists of the name of the step, the number of remaining candidates whose hyperbolicity is unknown, the numbers of candidate discarded within that step due to ineligibility, and the number of candidates kept to be dealt with later in Section 3.3.3.

For candidate generation, we used Regina’s utility command tricensus to generate all ideal triangulations consisting of $10$ tetrahedra except those that are obviously nonminimal or non-hyperbolic. More precisely, we obtained:

Due to Lemmas 4 and 5, the triangulations described in Theorem 7 contain all minimal ideal triangulations of cusped hyperbolic $3$-manifolds whose minimal ideal triangulations consist of $10$ tetrahedra. Note that triangulations representing nonorientable manifolds are also included and carried through the tests, though we will discard them in Section 3.3.

With the candidates generated, we proceed to determine their eligibilities. For an ideal triangulation to remain in the census, it must be both minimal and representing a cusped hyperbolic $3$-manifold. Hence an ideal triangulation is discarded immediately if it is certified to be nonminimal or representing a non-hyperbolic $3$-manifold. If an ideal triangulation is certified to represent a cusped hyperbolic $3$-manifold, we temporarily keep it, and decide whether it is minimal by checking if it represents a manifold that has already appeared in censuses with fewer tetrahedra, which we discuss in Section 3.3.

Four tests were used in Table 2, which we describe below:

Greedy nonminimality:

we use Regina’s highly efficient simplification algorithm [5] to try converting each candidate into an ideal triangulation with fewer tetrahedra, while representing the same $3$-manifold. If such a triangulation with fewer tetrahedra is found, then the candidate is nonminimal.

Exhaustive nonminimality:

given an integer parameter $h>0$, for each candidate with $n$ tetrahedra ($n=10$ in our settings), enumerate all triangulations that can be reached via $2$-$3$ and $3$-$2$ Pachner moves without ever exceeding $n+h$ tetrahedra. If a triangulation with fewer than $n$ tetrahedra is found during the process, the candidate is nonminimal.

Special surfaces:

for each candidate $𝒯$, enumerate all standard vertex normal surfaces in it, and check if any of them

• $\mathrm{■}$

  has positive Euler characteristic; or

• $\mathrm{■}$

  has zero Euler characteristic, is not a vertex link, and in addition

  • –

    is one-sided; or

  • –

    is two-sided, and if we cut $𝒯$ open along it then no component of the resulting space is a solid torus or solid Klein bottle.

If any such surface is found, then by Lemma 6, $𝒯$ is either nonminimal or does not represent a cusped hyperbolic $3$-manifold.

Certify hyperbolicity:

given an integer parameter $r>0$, for each candidate $𝒯$, we randomize it by doing random Pachner moves, and ask SnapPy to try finding a complete hyperbolic structure on $𝒯$, which uses an algorithm similar to HIKMOT’s [27], where affirmative answers are rigorous [24]. If the answer is not affirmative, we randomize $𝒯$ and try again, until an affirmative answer is achieved or the number of attempts reaches $r$.

We summarize the tests along with their core functions in Table 4 in [33, Appendix A.1].

As is presented in Table 2, for each candidate, an affirmative answer was given by one of the above four tests. A candidate was kept if and only if the affirmative answer was given by the certify hyperbolicity test, and was discarded otherwise. As such, we obtained $904,452$ triangulations certified to represent cusped hyperbolic $3$-manifolds at the bottom of Table 2, with all others discarded.

In this section, we discuss the procedure for step (iii) described in Section 3.1. A central tool for it, in both [7] and our method, is the canonical triangulation of cusped finite-volume hyperbolic $3$-manifolds.

For each of the $150,730$ isometry classes obtained in Section 3.3, we select a representative triangulation by the following procedure:

1. (1)

   If the verified canonical triangulation is in the isometry class, let it be the representative.

2. (2)

   Otherwise, consider the subset of triangulations that are geometric, which turns out to be never empty. If there is only one triangulation in the class, let it be the representative.

3. (3)

   If there are multiple triangulations in that subset, let the triangulation with the most self-isomorphisms, i.e. cellular homeomorphisms onto itself, be the representative.

4. (4)

   If there are multiple triangulations with the most self-isomorphisms, choose the one with the “fattest” tetrahedra, i.e. the one with the largest minimum imaginary part of tetrahedron shape parameters.

5. (5)

   If there are still multiple options left, choose the one with the smallest triangulation signature (as defined in [4]) under the lexicographical order.

With the representative triangulations chosen as above, we set their peripheral curves to be the shortest. We then ordered them by volume, named them as displayed in Table 1, saved and published the data in [34]. The data now comes automatically with SnapPy 3.3.

Note that verified canonical triangulations, by construction, are always geometric [22]. Hence due to the geometric requirement in step (2), we obtained

We remark that the above proposition also holds for all (orientable) censuses with fewer tetrahedra, which we checked using Burton’s data from [7].

As is discussed in Section 3.1, there are essentially three steps for creating a complete census. For the census of (orientable) cusped hyperbolic $3$-manifolds, step (ii) should not be a problem, as there are several other tests proposed in [7], but were not used when creating the $10$-tetrahedra census. For step (iii), the verified computation has a chance of failing only when the LLL algorithm is invoked, which happens only when a tilt that is extremely close to zero is encountered during computation. This essentially implies that the Epstein–Penner cellulation is not a triangulation, which is the case for only $986$ triangulations out of the $608918$ orientable candidates mentioned in Section 3.3.3, a frequency less than $0.2\%$. Therefore, it is promising that a combination of verified canonical triangulations with algebraic methods would suffice in the foreseeable future, where the former separates most manifolds, and the latter deals with the rest.

The main obstruction thus lives in step (i), where an exponential growth rate of the computational power is needed, with a much larger base compared to tabulating knots [39]. If one were to include nonorientable candidates during generation (as we did in Theorem 7), even though the algorithm established in [7] is already extremely efficient, it still takes approximately $2$ years to generate all candidates with $10$ tetrahedra in a single thread of $2.6$ GHz, about $50$ times longer than to generate all candidates with 9 tetrahedra; we accelerated the process by parallelizing it in $300$ threads. Thus, assuming the same growth rate, it will take approximately $100$ years to generate all candidates with $11$ tetrahedra with a single thread of $2.6$ GHz, and hence $4$ months in $300$ threads.

The time consumption and growth rate can both be lowered by excluding nonorientable candidates during generation, as half of identifications of each face will be excluded from the beginning. It takes $2$ months to enumerate all orientable candidates with $10$ tetrahedra in a single thread, about $25$ times longer than for $9$ tetrahedra. We successfully generated all orientable candidates with $11$ tetrahedra, which took about $6$ years in a single thread, giving Theorem 12. We have now completed the $11$-tetrahedra orientable census, with a preliminary SnapPy module available at [35], and full details to appear in a future publication.

Let $\overline{M}$ be a compact orientable $3$-manifold whose interior $M$ is a $1$-cusped hyperbolic $3$-manifold; in particular $\partial \overline{M}$ is a torus, and $M$ has finite volume. Dehn fillings on $M$ are characterized by slopes on $\partial \overline{M}$, which are unoriented isotopy classes of simple closed curves on the torus. For a slope $\alpha$, we denote by $M(\alpha )$ the closed manifold obtained by Dehn filling on $M$ so that $\alpha$ bounds a disk in the attached solid torus $D^2\times S^1$.

The census of exceptional Dehn fillings will consist of pairs $(M,\alpha )$, where $\alpha$ is exceptional.

By deleting a cusp neighborhood $N$ of $M$, we obtain a compact $3$-manifold $M\setminus N$ with torus boundary which is isotopic to $\overline{M}$. Slopes on $\partial \overline{M}$ can thus be carried by the isotopy onto $\partial (M\setminus N)\subset M$. By taking $N$ to be maximal, we define the length of a slope to be the minimum length of all curves in its isotopy class on $\partial (M\setminus N)$.

Agol [1] and Lackenby [31] independently proved that lengths of exceptional slopes cannot exceed $6$. It follows that there are only finitely many exceptional fillings on a $1$-cusped finite-volume hyperbolic $3$-manifold. Therefore it is possible to give complete lists of exceptional Dehn fillings on $1$-cusped hyperbolic $3$-manifolds in the censuses. For censuses of at most $9$ tetrahedra, this has been done by Dunfield [17]. For the $10$-tetrahedra census, we obtained:

Table 3 outlines the procedure for obtaining Theorem 14. There are $144,016$ $1$-cusped hyperbolic $3$-manifolds in the $10$-tetrahedra census. For each of them, we enumerated all slopes with length no more than $6$, obtaining a total of $800,447$ candidates for the exceptional census. As no duplicate slopes were enumerated for a given manifold, all candidates are distinct, and we only need to decide the hyperbolicity for each of them.

For each candidate $(M,\alpha )$, we consider its filled manifold $M(\alpha )$. Several rounds of certifying the hyperbolicity of $M(\alpha )$ with SnapPy reduced the remaining list by nearly a half. For Regina census lookup, we cut $M(\alpha )$ into connected sum summands and used regina.Census.lookup to try obtaining a name classifying it, which then tells us whether $M(\alpha )$ is hyperbolic or not. For those whose names were not obtained, we check whether they have fundamental normal surfaces which are essential $S^2$ or tori; if they do, they are not hyperbolic. Only $11$ candidates remain undetermined after this, and their filled manifolds were all identified as m135(1,3) by SnapPy. m135(1,3) is isometric to m412(1,-3)(-3,1) whose triangulation in SnapPy is certified to be geometric, hence m135(1,3) is hyperbolic.

Applying regina.Triangulation3.isSphere on fillings in Theorem 14 reveals that

For censuses of at most $9$ tetrahedra, similar work have been done by Callahan, Dean and Weeks [10], Champanerkar, Kofman and Patterson [12] and Dunfield [17].

Many conjectures and results mentioned in [17] can be tested and updated against the extended exceptional census. We plan to address these in a future publication.

Given an orientable cusped hyperbolic $3$-manifold $M$ with Betti number $b_1(M)=1$ and an $\mathrm{SL}(2,ℂ)$-lift $\tilde{\rho }$ of its holonomy representation $\rho :{\pi }_1(M)\to \mathrm{PSL}(2,ℂ)$, we consider its twisted Alexander polynomial ${\tau }_{2,\tilde{\rho }}(M)\in ℂ[t^{\pm 1}]$, defined by Dunfield, Friedl and Jackson [18, §2]. It was proved in [18] that

where $x(M)$ is the Thurston norm of a generator of $H^1(M;\mathbb{Z})$, and Equation 2 is an equality if $M$ fibers over $S^1$. Moreover, if $M$ fibers over $S^1$, then ${\tau }_{2,\tilde{\rho }}(M)$ is monic.

In [19], Dunfield, Garoufalidis and Yoon asked whether for any orientable cusped hyperbolic $3$-manifold with $b_1(M)=1$, there is at least one lift $\tilde{\rho }$ such that Equation 2 is an equality. An affirmative answer was then experimentally given for all such manifolds in the censuses of at most $9$ tetrahedra, a sample size of $59,068$ manifolds.

We extended the experimentation to the $10$-tetrahedra census. There are $143,919$ manifolds in the $10$-tetrahedra census with $b_1(M)=1$. By applying the code in [14], we were able to compute the Thurston norms for all except the following $21$ of them:

Out of the $21$ manifolds in Equation 3, the following $9$ were confirmed to not fiber over $S^1$, as their twisted Alexander polynomials are not monic:

For the remaining $143,898$ manifolds whose Thurston norms were successfully computed, we numerically computed their twisted Alexander polynomials of all $\mathrm{SL}(2,ℂ)$-lifts. After taking the maximum degrees, we found that Equation 2 is an equality for all of them.

The L-space conjecture asserts that for an irreducible $\mathbb{Q}$-homology $3$-sphere $Y$, the following are equivalent:

1. (a)

   ${\pi }_1(Y)$ is left-orderable;

2. (b)

   $Y$ has non-minimal Heegaard Floer homology;

3. (c)

   $Y$ admits a taut foliation.

The equivalence of (a) and (b) was proposed by Boyer, Gordon and Watson [3]. The equivalence of (b) and (c) was asked by Ozsváth and Szabó [38] after they proved that (c) implies (b), and upgraded to a conjecture by A. Juhász [30]. For $\mathbb{Z}$-homology spheres, Calegari and Dunfield [9] proved that (c) implies (a).

Dunfield [16] showed that the L-space conjecture holds on at least $62.2\%$ of $307,301$ hyperbolic $\mathbb{Q}$-homology $3$-spheres, obtained by Dehn fillings on orientable $1$-cusped hyperbolic $3$-manifolds in censuses of at most $9$ tetrahedra. It was conjectured in [16] that these are precisely the filled manifolds from those censuses with lengths of systoles being at least $0.2$.

While there are infinitely many hyperbolic $\mathbb{Q}$-homology $3$-spheres obtained by Dehn fillings on a finite set of orientable $1$-cusped hyperbolic $3$-manifolds, a finite list is obtained if we bound the list by the length of systoles, i.e. the shortest geodesics, from below, as the following result was shown by Hodgson and Kerckhoff [26]:

Here, the normalized length of $\alpha$ is the length of $\alpha$ divided by the area of $\partial (M\setminus N)$.

Similarly to Section 4.1, for the $1$-cusped hyperbolic $3$-manifolds in the $10$-tetrahedra census, by enumerating all slopes with normalized length no more than $7.515$, excluding all exceptional slopes, and filtering by the length of systole, we obtained a list of $1,899,974$ fillings containing all pairs $(M,\alpha )$ such that $M(\alpha )$ is a hyperbolic $\mathbb{Q}$-homology $3$-sphere with length of systole at least $0.163$. Therefore,

Changing $\mathbb{Q}$ to $\mathbb{Z}$ in Theorem 17, the number of pairs becomes $10,729$.

Note that, although the pairs $(M,\alpha )$ in Theorem 17 are all distinct, the filled manifolds $M(\alpha )$ are not; for instance, there are at least $22$ pairs in the list such that $M(\alpha )$ is identical to m007(3,1). It remains to group the pairs into distinct isometry classes by $M(\alpha )$, compare the grouped list against the list of $307,301$ manifolds in [16], and test the L-space conjecture on it. We wish to address these in a future publication.

An embedded surface $S$ in a hyperbolic $3$-manifold $M$ is called totally geodesic if every geodesic in $S$ is also a geodesic in $M$. The existence of closed totally geodesic surfaces is of particular interest.

An algorithm that is guaranteed to find all closed totally geodesic surfaces (up to isotopy) in a given orientable cusped finite-volume hyperbolic $3$-manifold was built and used to find several examples of cusped hyperbolic $3$-manifolds with closed totally geodesic surfaces by Basilio, Lee and Malionek [2]; however, none was found in the censuses of at most $9$ tetrahedra. By applying their algorithm to the $10$-tetrahedra census, we found that

In particular, among all orientable cusped hyperbolic $3$-manifolds containing closed totally geodesic surfaces, o10_143602 is the one whose minimal triangulations have the fewest tetrahedra; in other words, it is the simplest such example.