Hyperbolic graphs of small complexity

Table 1 gives a summary of our results. Up to complexity $4$ our census of hyperbolic graphs $(M,G)$ is complete and contains $45$ elements, consisting of 5 knots, 24 $\theta$-graphs, 13 handcuffs, and 3 distinct connected graphs with four vertices. The graph types occurring are shown in Figure 2. In complexity 5 we decided to rule out knot components, and we found 78 more hyperbolic graphs. Out of our 123 graphs, 36 lie in $S^{3}$.

Detailed information on all the 123 hyperbolic graphs up to complexity 5, including the volume and a description of the canonical decomposition, will be given in Section 5, while pictures of graphs up to complexity 4 will be shown in Section 7.

As shown in Table 1, there is a single hyperbolic graph of smallest complexity $c=1$. It is a handcuff graph in $S^{3}$, described in Figure 4 and Example 2.1. It is also the hyperbolic graph with vertices of least volume 3.663862377… This fact confirms the following relationships between complexity and hyperbolic geometry, which have already been verified for closed manifolds [32, 15], cusped manifolds [2, 3, 14, 15], and manifolds with arbitrary (geodesic) boundary [13, 26, 36, 11]:

1. (1)

   Objects having complexity zero are not hyperbolic.

2. (2)

   Among hyperbolic ones, the objects having lowest volume have the lowest complexity.

Note that complexity and volume may share the same first segments of hyperbolic objects (as they do) but are qualitatively different globally, because in general there are finitely many hyperbolic objects of bounded complexity, while infinitely many ones may have bounded volume thanks to Dehn surgery.

It may happen that $M\setminus G$ has a hyperbolic metric which completes to a manifold with compact totally geodesic boundary. In this case we say that $(M,G)$ is hyperbolic with geodesic boundary, which implies that $(M,G)$ is also hyperbolic (with parabolic meridians), but as mentioned above the converse is often false. By analyzing the graphs in Table 1, we have established the following:

The (0,1,2)-irreducible graphs of complexity 0 were detected by theoretical means, see Section 3. There are 3 knots (cores of Heegaard tori in $S^{3}$, $L(3,1)$ and ${\mathbb{P}}^{3}$) and the trivial $\theta$-graph in $S^{3}$, and they are all non-hyperbolic. In complexity $c=1,2$ we have classified all (0,1,2)-irreducible non-hyperbolic graphs, finding only 16 knots and two links. The same phenomenon happens for $c=3,4$, where we have shown that only knots and links are (0,1,2)-irreducible and non-hyperbolic. However we refrained from classifying them completely, confining ourselves to those in $S^{3}$ with $c=3$. Since our primary interest was in hyperbolic graphs, we decided to rule out knot components in complexity $5$, but quite interestingly we have found some non-hyperbolic examples in this case. Our results are summarized by Table 2 and the next statement:

A precise description of the knots, links and graphs appearing in Table 2 will be provided in Section 6.

We conclude this introduction by suggesting a few problems for further investigation.

1. (1)

   Enumerate the first few hyperbolic graphs with parabolic meridians in order of increasing hyperbolic volume.

2. (2)

   Enumerate the first few hyperbolic $3$-manifolds of finite volume with (compact or non-compact) geodesic boundary in order of increasing hyperbolic volume.

3. (3)

   Enumerate the first few closed hyperbolic 3-orbifolds in order of increasing complexity as defined in [41].

4. (4)

   Enumerate the first few closed hyperbolic 3-orbifolds in order of increasing hyperbolic volume.

5. (5)

   Determine the exact complexity of infinite families of knotted graphs, for example the torus knots in lens spaces (see Conjecture 6.4 below).

Note that Kojima and Miyamoto [26, 36] have already identified the lowest volume hyperbolic $3$-manifolds with compact and non-compact geodesic boundary. Perhaps the “Mom technology” introduced by Gabai, Meyerhoff and Milley [14, 15] may offer an approach to (1) and (2). Recent work of Martin with Gehring and Marshall [16, 29] has identified the lowest volume orientable hyperbolic 3-orbifold.

To help classify knotted graphs, we will study hyperbolic structures analogous to the compete hyperbolic structure on the complement of a knot or link. Given a graph $G$ in a closed orientable $3$-manifold $M$, let $N$ be the manifold obtained from $M\setminus G$ by removing an open regular neighbourhood of the vertex set of $G$. Thus $N$ is a non-compact $3$-manifold with boundary consisting of 3-punctured spheres, one corresponding to each vertex of $G$. Then we say that $(M,G)$ has a hyperbolic structure with parabolic meridians if $N$ admits a complete hyperbolic metric of finite volume with geodesic boundary (with toric and annular cusps). Equivalently, the double $D(N)$ of $N$ admits a complete hyperbolic metric of finite volume (with toric cusps). Such a hyperbolic structure on $N$ is unique by a standard argument using Mostow-Prasad rigidity [46] and Tollefson’s classification [49] of involutions with 2-dimensional fixed point set (see [47] and also [12]).

We next describe topological conditions for the existence of a hyperbolic structure with parabolic meridians. Let $X$ denote the graph exterior, i.e., the compact manifold obtained from $M$ by removing an open regular neighbourhood of the graph $G$. Then ${\partial}X$ is a disjoint union of pairs of pants (corresponding to the vertices of $G$) and a collection of annuli and tori $P\subset X$ (corresponding to the edges and knots in $G$). Thurston’s hyperbolization theorem for pared $3$-manifolds [37, 23] implies the following:

The conditions for hyperbolicity simplify considerably when $(M,G)$ is $(0,1,2)$-irreducible, as defined in the introduction. To elucidate the notion, we say that $(M,G)$ is:

• •

  0-irreducible if every 2-sphere in $M$ disjoint from $G$ bounds a 3-ball in $M$ disjoint from $G$;

• •

  1-irreducible if there exists no 2-sphere in $M$ meeting $G$ transversely in a single point;

• •

  2-irreducible if every 2-sphere in $M$ meeting $G$ transversely in two points bounds a ball in $M$ that intersects $G$ in a single unknotted arc.

Then a graph is $(0,1,2)$-irreducible if it is $i$-irreducible for $i=0,1,2$.

Let $(M,G)$ be a graph, and let $X$ denote the graph exterior as above. Let us define $Y$ as the manifold obtained by mirroring $X$ in its non-toric boundary components, so $Y$ is either closed or bounded by tori. Then $X$ minus its toric boundary components has a hyperbolic structure with totally geodesic boundary if and only if the interior of $Y$ has a complete hyperbolic structure. By Thurston’s hyperbolization theorem [37, 23] and Mostow-Prasad rigidity (see [47, p. 14]) we then have:

Comparing Theorems 2.2 and 2.6 one easily sees that if $X$ minus its toric boundary components admits a hyperbolic structure with geodesic boundary, then $(M,G)$ admits a hyperbolic structure with parabolic meridians. The converse is however false, as most of the pairs $(M,G)$ described below show.

One of the initial motivations of our work was the study of hyperbolic $3$-orbifolds, but the analysis of graphs turned out to be interesting enough by itself, so we decided to leave orbifolds for the future. However we mention them briefly here.

Given a trivalent graph $G$ in a closed $3$-manifold $M$, we obtain an orbifold $Q$ associated to $(M,G)$ by attaching an integer label $n_{e}\geqslant 2$ to each edge or circle $e$ of $G$. Note that we do not impose any restrictions on the labels $(p,q,r)$ of the edges incident to a vertex $v$, so from a topological viewpoint $v$ gives rise either to an interior point of $Q$ (if $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}>1$) or to a boundary component of $Q$ — a 2-orbifold of type $S^{2}(p,q,r)$.

We will say that $Q$ is hyperbolic if $M\setminus G$ admits an incomplete hyperbolic metric whose completion has a cone angle $\frac{2\pi}{n_{e}}$ along each edge or circle $e$ in $G$. Depending on whether $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}-1$ is positive, zero or negative, a vertex with incoming labels $(p,q,r)$ gives rise to an interior point of $Q$ to which the singular metric extends, to a cusp of $Q$, or to a totally geodesic boundary component of $Q$.

The main connections between orbifold hyperbolic structures and those we deal with in this paper are as follows:

• •

  If $(M,G)$ has a hyperbolic orbifold structure for some choice of labels $n_{e}$, then $(M,G)$ admits a hyperbolic structure with parabolic meridians.

• •

  If $(M,G)$ admits a hyperbolic structure with parabolic meridians then the corresponding orbifolds are hyperbolic provided all labels are sufficiently large; moreover the structure with parabolic meridians can be regarded as the limit of the orbifold hyperbolic structures as all labels tend to infinity.

The first assertion follows from Theorem 2.2 by topological arguments only (see [1, Prop. 6.1]), while the second one is a consequence of Thurston’s hyperbolic Dehn surgery theorem (see [1, 7] for details).

As already mentioned, the hyperbolic structures and related invariants on the 123 pairs of our census have been obtained using the computer program Orb [19]. More details on this program will be provided below, but we outline here the underlying theoretical idea (due to Thurston [46]) of the algorithmic construction of a hyperbolic structure with geodesic boundary on a pared manifold $(X,P)$, where $X$ is compact but not closed and $P$ is a collection of tori and annuli on $\partial X$.

The starting point is a (suitably defined) ideal triangulation of $(X,P)$, namely a realization of $(X,P)$ as a gluing of generalized ideal tetrahedra. Each of these is a tetrahedron with its vertices removed and, depending on its position with respect to $\partial X$ and $P$, perhaps entire edges and/or open regular neighbourhoods of vertices also removed. The next step is to choose a realization of each of these tetrahedra as a geodesic generalized ideal tetrahedron in hyperbolic $3$-space. These realizations are parameterized by certain moduli, and the condition that the hyperbolic structures on the individual tetrahedra match up to give a hyperbolic structure on $(X,P)$ translates into equations in the moduli. The algorithm then consists of changing the initial moduli using Newton’s method until the (unique) solution of the equations is found.

When $M$ is closed one can search for its hyperbolic structure using a similar method, starting from a decomposition of $M$ into compact tetrahedra [4].

Whenever a hyperbolic manifold $X$ is not closed, it admits a canonical decomposition into geodesic hyperbolic polyhedra, which allows one to very efficiently compute its symmetry group and compare it for equality with another such manifold. The decomposition was defined by Epstein and Penner [9] when $\partial X=\emptyset$ but $X$ has cusps, and by Kojima [24, 25] when $\partial X\neq\emptyset$. We will now briefly outline the latter construction.

Begin with the geodesic boundary components of $X$ and very small horospherical cross sections of any torus cusps of $X$, and expand these surfaces at the same rate until they bump to give a 2-complex (the cut locus of the initial boundary surfaces). Then dual to this complex is the Kojima canonical decomposition of $X$ into generalized ideal hyperbolic polyhedra. This is independent of the choice of horosphere cross sections provided they are chosen sufficiently small, and gives a complete topological invariant of the manifold.

Thus two finite volume hyperbolic $3$-manifolds with geodesic boundary are isometric (or, equivalently, homeomorphic) if and only if their Kojima canonical decompositions are combinatorially the same; and the symmetry group of isometries of such a manifold is the group of combinatorial automorphisms of the canonical decomposition. Similarly, two graphs admitting hyperbolic structures with parabolic meridians are equivalent if and only if there is a combinatorial isomorphism between their canonical decompositions taking meridians to meridians; and the group of symmetries of such a graph is the group of combinatorial automorphisms of the canonical decomposition taking meridians to meridians.

Let us first note that a hyperbolic structure on an orientable $3$-manifold without boundary corresponds to a realization of the manifold as the quotient of hyperbolic space ${\mathbb{H}}^{3}$ under the action of a discrete group $\Gamma$ of orientation-preserving isometries of ${\mathbb{H}}^{3}$. If the manifold has boundary, ${\mathbb{H}}^{3}$ should be replaced by a $\Gamma$-invariant intersection of closed half-spaces in ${\mathbb{H}}^{3}$. Moreover for any given hyperbolic $3$-manifold, the group $\Gamma$ is well-defined up to conjugation within the full group of orientation-preserving isometries of ${\mathbb{H}}^{3}$, which is isomorphic to $\mbox{PSL}(2,{\mathbb{C}})$.

If $\Gamma$ is a discrete subgroup of $\mbox{PSL}(2,{\mathbb{C}})$, then the invariant trace field $k(\Gamma)\subset{\mathbb{C}}$ is the field generated by the traces of the elements of $\Gamma^{(2)}=\{\gamma^{2}\mid\gamma\in\Gamma\}$ lifted to $\mbox{SL}(2,{\mathbb{C}})$. This is a commensurability invariant of $\Gamma$ (unchanged if $\Gamma$ is replaced by a finite index subgroup). Further, if ${\mathbb{H}}^{3}/\Gamma$ has finite volume then it follows from Mostow-Prasad rigidity that $k(\Gamma)$ is a number field, i.e., a finite degree extension of the rational numbers ${\mathbb{Q}}$. (See [28] for an excellent discussion and proofs.)

If a trivalent graph $(M,G)$ admits a hyperbolic structure $N$ with parabolic meridians, then $N$ is the convex hull of ${\mathbb{H}}^{3}/\Gamma$ where $\Gamma$ is a discrete subgroup of $\mbox{PSL}(2,{\mathbb{C}})$. Thus $k(\Gamma)$ is an invariant of $(M,G)$. Now the double $D(N)$ (defined at the start of Subsection 2.1) has the form ${\mathbb{H}}^{3}/\Gamma_{1}$, where $\Gamma_{1}$ is a Kleinian group containing $\Gamma$. Since $D(N)$ is hyperbolic with finite volume, $k(\Gamma_{1})$ is an algebraic number field. Hence the subfield $k(\Gamma)$ is also an algebraic number field. We compute this by combining Orb with a modified version of Oliver Goodman’s program Snap ([17]).

Snap begins with generators and relations for $\Gamma$, and a numerical approximation to $\Gamma$ provided by Orb. It first refines this using Newton’s method to obtain a high precision numerical approximation to $\Gamma$, and then tries to find exact descriptions of matrix entries and their traces as algebraic numbers using the LLL-algorithm. Finally Snap verifies that we have an exact representation of $\Gamma$ by checking that the relations for $\Gamma$ are satisfied using exact calculations in a number field, and computes the invariant trace field $k(\Gamma)$ and associated algebraic invariants. (See [8] for a detailed description of Snap.)

To proceed with the key notions and results we recall a definition given in the Introduction. We call simple¹¹1In [32] such a polyhedron was originally called almost simple, while the term simple was employed for almost special polyhedra, see Subsection 3.2. a compact polyhedron $P$ (in the PL sense [44]) such that the link of each point is a subset of the 1-skeleton of the tetrahedron. We denote by $V(P)$ the set of points of $P$ having the whole 1-skeleton of the tetrahedron as a link, and we note that $V(P)$ is a finite set.

It is very easy to see (and it will follow from the duality with triangulations in Proposition 3.2) that each $(M,G)$ admits simple spines. Therefore the complexity of $(M,G)$, that we define as

is a finite number.

To illustrate the relation between spines and triangulations, we need to introduce two subsequent refinements of the notion of simple polyhedron. We will say that $P$ is almost-special if it is a compact polyhedron and each of its points has one of the following sets as a link:

1. (1)

   The $1$-skeleton of the tetrahedron with two open opposite edges removed (a circle);

2. (2)

   The $1$-skeleton of the tetrahedron with one open edge removed (a circle with a diameter);

3. (3)

   The $1$-skeleton of the tetrahedron (a circle with three radii).

The corresponding local structure of an almost-special polyhedron is shown in Figure 6.

Besides the set $V(P)$ of vertices already introduced above for simple polyhedra, we can define for an almost-special $P$ the singular set, given by the non-surface points and denoted by $S(P)$. We remark that $S(P)$ is a 4-valent graph with vertex set $V(P)$. Note also that if $P$ is an almost-special spine of $(M,G)$, by the transversality assumption, $G$ intersects $P$ away from $S(P)$.

An almost-special polyhedron $P$ is called special if $P\setminus S(P)$ is a union of open discs and $S(P)\setminus V(P)$ is a union of open segments. A special spine of a graph $(M,G)$ is a simple spine which, in addition, is a special polyhedron.

The following result, which refers to the case of manifolds without graphs embedded in them, has been known for a long time. We point out that we use the term triangulation for a (closed, connected, orientable) $3$-manifold $M$ in a generalized (not strictly PL [44]) sense. Namely, we mean a realization of $M$ as a simplicial pairing between the faces of a finite union of tetrahedra, i.e., we allow multiple and self-adjacencies between tetrahedra.

We now turn to graphs $(M,G)$, and we define a triangulation of $(M,G)$ to be a (generalized) triangulation ${\mathcal{T}}$ of $M$ which contains $G$ as a subset of its 1-skeleton. We will further say that ${\mathcal{T}}$ is efficient if it has precisely one vertex at each vertex of $G$, one on each knot component of $G$, and no other vertices.

The following easy result shows that under suitable conditions Proposition 3.2 has a refinement to graphs:

A simple spine $P$ of a graph $(M,G)$ is called minimal if it has $c(M,G)$ vertices and no subset of $P$ is also a spine of $(M,G)$. The success of the strategy based on complexity theory (as outlined at the beginning of this section) for the enumeration of hyperbolic graphs depends on the next three results. They require the concept of $(0,1,2)$-irreducibility defined in the introduction. The first one is part of Theorem 2.4, the next two easily follow from [41, Theorem 2.6].

The following result was used for the enumeration of candidate triangulations of $(0,1,2)$-irreducible graphs, as explained in more detail in the next section:

Matveev’s complexity [32] is defined for every compact $3$-manifold, with or without boundary. The complement $X$ of an open regular neighbourhood of a graph $G$ in a closed $3$-manifold $M$ therefore has a complexity, which is related to $c(M,G)$ as follows.

Thanks to Theorem 3.6 and the other results stated in the previous section, the enumeration of $(0,1,2)$-irreducible graphs of complexity $n$ can be performed by listing all efficient triangulations with $n$ tetrahedra satisfying some minimality criteria. This was done via a separate program, written in Haskell [18], which suitably adapts the strategy already used in similar censuses (e.g. [30, 35]).

A triangulation of a graph $(M,G)$ can be encoded as a triangulation of $M$ with some marked edges. A triangulation here is just a gluing of tetrahedra, which can be described via a connected $4$-valent graph (the incidence graph of the gluing) having a label on each edge encoding how the corresponding triangular faces are identified (there are $3!=6$ possibilities).

A first count gives $c_{n}\cdot 6^{2n}=c_{n}\cdot 36^{n}$ triangulations to check, where $c_{n}$ is the number of $4$-valent graphs with $n$ vertices (and $2n$ edges), shown in Table 3. On each triangulation there are $2^{e}=2^{n+v}$ distinct markings of edges, where $e$ is the number of edges and $v$ is the number of vertices in the triangulation of $M$. Since there are at least $2$ triangles in the link of each vertex, $v\leqslant 2n$, and $e\leqslant 3n$. There are therefore up to $c_{n}\cdot 36^{n}\cdot 2^{3n}=c_{n}\cdot 288^{n}$ marked triangulations to check. This number is already too big for $n=3$, so in order to simplify the problem we used some tricks.

We are only interested in orientable manifolds $M$. We can therefore orient each tetrahedron and require the identifications of faces to be orientation-reversing. This reduces the number of possible labels on edges from $6$ to $3$, and the number of triangulations to $c_{n}^{\prime}\cdot 3^{2n}=c_{n}^{\prime}\cdot 9^{n}$, where $c_{n}^{\prime}$ is the number of $4$-valent graphs with “oriented” vertices: each vertex has a fixed parity of orderings of the incident edges. For a fixed $4$-valent graph $G$ with $n$ vertices, the vertices can be oriented in $2^{n}$ different ways, but up to the symmetries of $G$ the number of distinct orientations typically turns out to be very small. This explains why $c_{n}^{\prime}$ is actually much less than $2^{n}\cdot c_{n}$, as shown in the table.

We selected from the resulting list of triangulations only those yielding closed manifolds. Finally, on each triangulation we a priori had $2^{e}$ distinct markings on edges to analyze. Proposition 3.7 was used to discard many of these: in a triangulation dual to a minimal spine an edge incident to at most $3$ distinct tetrahedra is necessarily marked. It remained then to check which markings give rise to efficient triangulations.

Hyperbolic structures were computed using the program Orb written by Damian Heard [20, 19]. This program builds on ideas of Thurston, Weeks, Casson and others to find hyperbolic structures and associated geometric invariants for a large class of 3-dimensional manifolds and orbifolds. The program begins with a triangulation of the space with the singular locus or graph contained in the 1-skeleton and tries to find shapes of generalized hyperbolic tetrahedra (with vertices inside, on, or outside the sphere at infinity) which fit together to give a hyperbolic structure.

The generalized hyperbolic tetrahedra are described by using one parameter for each edge in the triangulation. For a general tetrahedron a lift to Minkowski space is chosen, then the parameters are Minkowski inner products of the vertex positions. For compact tetrahedra, each parameter is just the hyperbolic cosine of the edge length. For each ideal vertex the lift to Minkowski space determines a horosphere centred at the vertex; for each hyperideal vertex a geodesic plane orthogonal to the incident faces is determined. Then the edge parameters are simple functions of the hyperbolic distances between these surfaces.

Given the edge parameters, all dihedral angles of the tetrahedra are determined. Moreover the parameters give a global hyperbolic structure if and only if the sum of the dihedral angles around each edge is $2\pi$ (or the desired cone angle, in the orbifold case). This gives a system of equations that Orb solves numerically using Newton’s method, starting with suitable regular generalized tetrahedra as the initial guess.

Once a hyperbolic structure is found, Orb can compute many geometric invariants including volumes, the Kojima canonical decompositions, and symmetry groups. This uses methods based on ideas of Weeks [52], Ushijima [50] and Frigerio-Petronio [12], too complicated to be reproduced here.

After computing hyperbolic structures numerically using Orb, we checked the correctness of the results by using Jeff Weeks’ program SnapPea [51] to calculate complete hyperbolic structures on the manifolds with torus cusps obtained by doubling along all 3-punctured sphere boundary components.

Finally, we verified the results by using Oliver Goodman’s program Snap [17, 8] to find exact hyperbolic structures. This provides a proof that the hyperbolic structures are correct and allows us to compute associated arithmetic invariants (including invariant trace fields), as already mentioned in Subsection 2.6 above.

Many knots and links in the census turned out to be torus links in lens spaces, see Subsection 6.1 below. From $c=3$, we then decided to rule out the non-hyperbolic knots and links from our census (except for those in $S^{3}$ at $c=3$); this helped a lot in simplifying the classification. Many non-hyperbolic knots and links were easily identified by the following criterion:

The remaining knots and links were shown to be non-hyperbolic by examining their fundamental groups with the help of the following observations.

Among the knots and links up to complexity 4 for which Orb did not find a hyperbolic structure, all but one of the complements had a fundamental group with presentation of the form $\langle a,b\mid[a^{n},b^{m}]=1\rangle$, or $\langle a,b\mid a^{n}(a^{p}b^{q})^{k}=1\rangle$. These all correspond to non-hyperbolic links by the Lemmas above. The one remaining knot had a presentation as in part (ii) of Lemma 4.3, so is also non-hyperbolic.

For graphs with at least one vertex, we first eliminated all triangulations whose dual spines had non-minimal complexity hence were either reducible or occurred earlier in our list. This left a handful of examples for which Orb failed to find a hyperbolic structure. These were first examined using Jeff Weeks’ program SnapPea, by constructing triangulations of the manifolds with torus cusps obtained by doubling along the 3-punctured sphere boundary components. We used SnapPea’s “splitting” function to look for incompressible Klein bottles and tori in the doubles. This suggested that incompressible Klein bottles were present in the original graph complements. We then verified this and showed that these examples were indeed non-hyperbolic by theoretical means, as explained below in Section 6.

For future reference, we have chosen a name for each of the graphs we have found. The name has the form

where $n$ is the number of vertices of the graph, $g$ is a string describing the abstract graph type, $c$ is the complexity, and $i$ is an index (starting from $1$ for any given $ng\_\,c$). We have found in our hyperbolic census only 6 graph types, described above in Figure 2, so a string of one letter only (or the empty string, for knots) was sufficient to identify them. For graphs with 2 vertices, the letters $t$ and $h$ were suggested by the common names “$\theta$-graph” and “handcuffs”. The choice of letters was arbitrary for graphs with $4$ vertices.

We will give separate tables for $\theta$-graphs, handcuffs, $4$-vertex graphs, and knots. Within each table, graphs are always arranged in increasing order of their hyperbolic volumes. For graphs having vertices, the columns of the tables respectively contain:

1. (1)

   The name of the graph $(M,G)$.

2. (2)

   The volume of the hyperbolic structure with parabolic meridians on $M\setminus G$.

3. (3)

   A description of the cells of the Kojima canonical decomposition for this structure. When all these cells are tetrahedra we simply indicate their number, otherwise we add an asterisk in the table and provide additional information separately.

4. (4)

   The symmetry group of $(M,G)$, with $D_{n}$ denoting the dihedral group with $2n$ elements.

5. (5)

   Whether $(M,G)$ is chiral (c) or amphichiral (a).

6. (6)

   The name of the underlying space $M$. This is almost always a lens space; otherwise it is a Seifert fibred space which we describe in the usual way (as in [34, p. 406]).

7. (7)-(9)

   The degree, signature and discriminant of the invariant trace field. Details of minimal polynomials for the fields are available on the web at www.ms.unimelb.edu.au/˜snap/knotted_graphs.html.

8. (10)

   Whether all traces of group elements are algebraic integers.

9. (11)

   Whether the group is arithmetic (after doubling to obtain a finite covolume group).