## Abstract.

By a regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we prove that a regular tessellations has to be infinite volume if its fundamental group is generated by peripheral curves only. This shows that there are at least 19 and at most 21 link complements that are regular tessellations (computer experiments suggest that at least one of the two remaining cases likely fails to be a link complement, but so far, we have no proof). In particular, we complete the classification of all principal congruence link complements given in Baker and Reid for the cases of discriminant and . We only describe the manifolds arising as complements of links here, with a future publication “Regular Tessellation Links” giving explicit pictures of these links.

## 1. Introduction

According to [AR92], tessellations by Platonic solids “have played a significant role in the exploration and exposition of 3-dimensional geometries and topology,” and we refer the reader to Aitchison and Rubinstein’s excellent introduction for examples. Existing literature has investigated the tessellations by ideal (and thus hyperbolic) Platonic solids that can occur as knot and link complements. Following [Cox56, Cox73], we know that there is an ideal tessellation of hyperbolic 3-space by the regular ideal tetrahedron, octahedron, cube, and dodecahedron, but not the icosahedron. It is known that each of the first four Platonic solids also tessellates some knot or link complement. An example for the tetrahedron is the figure-eight knot, and for the octahedron the Borromean rings [Thu97]. Aitchison and Rubinstein completed this work by constructing cubical links and dodecahedral knots [AR92]. Note that for some solids, the literature gives knots, but for others, only links. In fact, the figure-eight knot and the two dodecahedral knots are the only three knots whose complements are tessellated by an ideal Platonic solid. This follows from Reid’s result that the figure-eight knot is the unique arithmetic knot [Rei91] and Hoffman’s analysis of the nonarithmetic dodecahedral case [Hof14].

However, there are infinitely many link complements tessellated by ideal Platonic solids. To see this, pick some link that has an unknotted component and whose complement is tessellated by an ideal Platonic solid, for example, the Borromean rings. Then, construct an arbitrarily large cyclic cover of the link branched over the chosen unknotted component. If we require as natural symmetry condition that the link complement be a regular ideal tessellation, then the number becomes finite again, and it is a natural question and the goal of this paper to classify them. Note that all manifolds considered here are orientable but that we allow chiral regular tessellations of a manifold, so a tessellation is defined to be regular if each flag consisting of a solid, an adjacent face and an edge adjacent to the face can be taken to any other flag through an isometry. When we say “regular tessellation” we mean any hyperbolic manifold together with a tessellation fulfilling this condition. This generalizes the traditional use of that term that assumed that the underlying manifold is , , or . For the tetrahedron, a tessellation is a regular if and only if the underlying hyperbolic manifold is “maximally symmetric” in the sense that no other manifold has more orientation-preserving symmetries per volume (this follows from [Mey86] where the minimum volume orbifold was determined).

### 1.1. Examples of Known Regular Tessellations

Ideal regular tessellations by Platonic solids arise in various contexts. If we allow orbifolds for a moment, there is a particularly easy example: the boundary of a 4-simplex is a regular tessellation by 5 regular ideal hyperbolic tetrahedra, as shown on the left in Figure 1. To obtain a regular tessellation of a cusped hyperbolic manifold, take the unique manifold double-cover of the orbifold. The result is the complement of the minimally twisted 5-component chain link L10n113, which conjecturally has the smallest volume among the 5-cusped orientable hyperbolic manifolds. Its Dehn fillings have been extensively studied: the exceptional ones were recently classified in [MPR14], and the remaining Dehn fillings yield almost every manifold of the cusped Callahan-Hildebrand-Weeks census [CHW99] and thus also of the closed Hodgson-Weeks census [HW94], a remarkable property also featured in the experiments of [DT03] on the virtual Haken conjecture.

Another example of a regular tessellation with an easy combinatorial description comes from chessboard complexes defined in [Zie94, BLVŽ94]:

###### Definition 1.1.

The chessboard complex is a simplicial complex consisting of a -simplex for every non-taking configuration of rooks (“that is, no two rooks on the same row or column”) on the chessboard. A face of a -simplex is identified with the -simplex corresponding to the respective subset of rooks.

The chessboard complex happens to yield a regular tessellation by 120 hyperbolic ideal tetrahedra [Epp] (namely ). To see this, look at the and the chessboard complex: The chessboard complex is a 6-cycle. The 6-cycle is the link of a vertex in the chessboard complex that is a torus with 24 triangles. This torus is a link of a vertex in the chessboard complex. Hence, there are 6 tetrahedra meeting at an edge and removing the vertices yields a 3-manifold with toroidal cusps.

The last example we give is the Thurston congruence link, an 8-component link whose complement admits a regular tessellation by 28 hyperbolic ideal tetrahedra with a fascinating connection to the Klein quartic, the complex projective curve defined by . In [Thu98], Thurston noticed that the cusp neighborhoods in the complement of the minimally twisted 7-component chain link appear to be close to a regular pattern and that drilling a geodesic “crystallizes” the hyperbolic manifold so that it admits a regular tessellation by 28 ideal tetrahedra (namely ). He gave a picture of the resulting 8-component link and noticed a fascinating connection with the Klein quartic: they share the same orientation-preserving symmetry group , the unique simple group of order 168. In fact, the 2-skeleton of the above regular tessellation forms an immersed punctured Klein quartic [Ago].

### 1.2. Arithmetic and Congruence Links

The Thurston congruence link is also an example of a congruence link and thus illustrates the connection of the classification result here to arithmetic and congruence links. Motivated by Thurston’s question 19 in [Thu82] noting the “special beauty” of these manifolds, Reid and Baker have been classifying small congruence links, e.g., [BR14]. For a subclass of congruence links, namely the principal congruence links for discriminant and , the main theorem here allows a complete classification. Let where is the ring of integers in the imaginary quadratic number field . Recall that the principal congruence manifold (denoted here by , see Section 5) of level is the quotient of by all matrices in that are congruent to the identity matrix modulo the ideal . For discriminant and , a principal congruence manifold admits a regular tessellation by ideal tetrahedra, respectively, octahedra. Thus, we can apply the main theorem to list all and that are link complements in Corollary 5.3.

### 1.3. Remarks

It should be remarked that although the Borromean rings and the Whitehead link can be tessellated by octahedra and the alternating 4-component chain link by cubes, these tessellations are not regular as the smallest regular tessellation link (smallest in terms of both volume and number of components) has 5 components. In particular, no knot complement is a regular tessellation.

Furthermore, note that there is a subtle difference between classifying regular tessellation link complements and regular tessellation links. Whereas the number of regular tessellation link complements is finite, the number of regular tessellation links is, a priori, not finite as a link is in general not determined by its complement (as pointed out in [BR14]). In fact, twisting along a disc spanned by a component of the link in Figure 1 yields an infinite family of regular tessellation links with the same complement. Thus, we only classify link complements here, not links.

## 2. Main Theorem and Overview

Consider the regular ideal tessellations of by ideal Platonic solids (see [AR92]): tessellated by tetrahedra, by octahedra, by cubes, and by dodecahedra. We use the upper half space model of and identify with so that and act by Möbius transformations and . For normalization, we move each of the above regular ideal tessellations in such that there is a face with three consecutive vertices at the points , , and . Let be the orientation-preserving isometries of such a regular tessellation of .

###### Definition 2.1.

A manifold is a regular tessellation of type if it is a quotient by a torsion-free normal subgroup of .

Let be a number of the form where and . For , we write and, for , where .

###### Definition 2.2.

The universal regular tessellation of cusp modulus is the quotient

 U{p,q,r}z=H3U{p,q,r}z

where denotes the normal closure of in , i.e., the smallest normal subgroup of containing .

The groups were also called “stabilizer of infinity in the principal congruence subgroup” in [BR14]. The quotient space can actually be an orbifold. It can also be infinite-volume. We explain the regular tessellation structure and the universal property of in detail in Section 4 and only remark here that multiplication of by leaves the universal regular tessellation unchanged and complex conjugation only reflects it. Thus, for classification purposes, we only have to enumerate those that are in canonical form defined as follows:

###### Definition 2.3.

Let . If , we say that is in canonical form.

We can now state the main theorem:

###### Theorem 2.4.

If is a link complement admitting a regular tessellation, then it must be a finite-volume manifold universal regular tessellation. All finite-volume universal regular tessellations are listed in Table 1, so there are 20 or 21 non-homeomorphic such manifolds. For every manifold case not marked with in the table, the universal regular tessellation is known to be a link complement. Thus there are 19 to 21 regular tessellation link complements.

Note that there are two unsettled cases and . In the first case, it is unclear whether the universal regular tessellation is finite-volume. Computer experiments strongly indicate that it is indeed infinite volume and thus not a link complement; see also the discussion in Section 12. In the second case, we know that the universal regular tessellation is finite-volume, but could neither prove nor disprove that it is a link complement.

This leaves the question unsettled whether every finite volume manifold universal regular tessellation is a link complement. Baker and Reid posed a more general formulation of this question in [BR14]. We will discuss this further in the discussion section.

For some cases, a link with complement could be found easily or has been explicitly constructed: is shown in Figure 1, the complement of the Thurston congruence link [Thu98] is , links for and have been constructed in the author’s PhD thesis [Gör11]. Finally, the complement of the minimally twisted 6-component chain link turns out to be .

The rest of the paper is organized as follows: We first develop the theory of regular tessellations in Section 3 and 4. An important invariant is the cusp modulus describing the regular tessellation structure when restricting to a cusp neighborhood. The largest tessellation and only potential link complement among all tessellations of given cusp modulus is the universal regular tessellation.

Section 5 classifies all regular tessellations for small cusp modulus and all principal congruence link complements for discriminant and .

Section 6 introduces the central algorithm in this paper: the construction of the universal regular tessellation. This has been implemented in python (see http://www.unhyperbolic.org/regTess/) and Section 7 describes the details of the implementation as well as examples of how to use the software to obtain the results in Table 1 and Section 5.

The rest of the paper is devoted to proving the main theorem which means proving that the algorithm to construct the universal regular tessellation never terminates for a case not listed in Table 1. Section 8 does this for large cusp modulus. Section 9 introduces cuspidal homology to construct cuspidal covers. This is used in Section 10 to prove infinite universal regular tessellations in the remaining cases.

## 3. Regular Tessellations of the Torus

Let be the ring of integers of the imaginary quadratic number field of discriminant . Here, we will focus on the Eisenstein integers with and the Gaussian integers . Let denote the generator , respectively, of each of these two rings which are principal ideal domains so every ideal is of the form for some which is determined by the ideal up to a unit .

Draw a line segment between each pair of points in that have unit distance. The result is a regular tessellation of type for , respectively, for .

###### Definition 3.1.

Given , let be the triangulation of the torus obtained as quotient of the above regular tessellation by the action of the elements in the ideal by translations. The dual tessellation of the torus is denoted by .

Recall that every regular tessellation of the torus is of the form or . An example is given in Figure 2. Given an oriented regular tessellation of a torus, the classifying the tessellation is determined up to multiplication by a unit . Note that is the mirror image of . Hence, given an unoriented regular tessellation of the torus, is only defined up to multiplication by a unit and complex conjugation. Furthermore, is chiral if and only if .

## 4. Cusp Modulus

Recall that we normalized the regular tessellations such that there is a face with three consecutive vertices at the points , and , see Figure 3. Take a horosphere about that is a high enough plane parallel to . A solid of an ideal regular hyperbolic tessellations intersects in a regular -gon and thus the regular tessellation of induces a regular tessellation of type on with vertices at where . The orientation-preserving isometries of the tessellation on are given by the upper triangular matrices with coefficients in with upper unit-triangular matrices corresponding to translations. In fact, coincides with the natural -extension of a Bianchi group in two cases: is given by and by . One of the other two cases, , is also arithmetic because it is commensurable with but does neither cover it nor is covered by it. However, is not arithmetic. As group, each is the orientation-preserving index-2 subgroup of a Coxeter reflection group.

Now consider a regular tessellation of a finite-volume oriented cusped hyperbolic manifold and of type . Similarly, there is an induced oriented tessellation of the boundary of a cusp neighborhood for each cusp, this time on a torus. The tessellation of the 3-manifold being regular also implies that the induced tessellation is regular and the same for each cusp. Thus, the induced oriented regular tessellation is given by some . Thus, we obtain an invariant of defined up to multiplication by a unit which we call the cusp modulus. Note that for the non-arithmetic case , the cusp modulus is still an element in but that is not the invariant trace field.

###### Remark 4.1.

The regular tessellation structure is also equal to the canonical cell decomposition of the underlying hyperbolic manifold [EP88]. In particular, a hyperbolic 3-manifold admits at most one regular tessellation structure and the cusp modulus is an invariant of the underlying hyperbolic 3-manifold admitting a regular tessellation. Note that this is in general false for non-regular tessellations by Platonic solids. For example, the manifold underlying the regular tessellation of ten tetrahedra also admits several different tessellations by two regular cubes but none of the cubical tessellations is regular and thus equal to the canonical cell decomposition ( is a double cover of the orbifold in Figure 1. This orbifold was named in [NR92] and shown to decompose into a cube).

We can also characterize tessellations algebraically. A manifold has a tessellation induced from the tessellation of if is a torsion-free subgroup of . The induced tessellation on is a regular tessellation if furthermore is normal in (this is how it was defined earlier in Definiton 2.1). To see this, recall that a tessellation is regular if its orientation-preserving symmetry group acts transitively on all flags. acts freely and transitively on all flags of the tessellation of . Thus a manifold quotient is a regular tessellation if every symmetry of descends to a symmetry of which is equivalent to being normal in .

Let

 pz=(1z01),P={pw:w∈C}% andPz={pw:w∈⟨z⟩}.
###### Definition 4.2.

Let be a cusped orientable 3-manifold such that where is a torsion-free normal subgroup of . Then is the cusp modulus of if .

Note that in this definition, the cusp modulus is defined up to a multiplication by a unit again. To see this, look at the group . If and , then and where because is normal. Thus, the off-diagonal entries of the matrices in form an ideal and the ideal is generated by an element determined up to .

This definition of cusp modulus also matches the earlier geometric definition of cusp modulus: the group acts on the regular tessellation on the horosphere by translations and the resulting quotient is the tessellated torus .

Recall the universal regular tessellation from Definition 2.2. If it is a manifold, it is a regular tessellation by definition and it always has the following universal property:

###### Lemma 4.3.

Let be a (not necessarily finite volume) regular tessellation with cusp modulus . Then there is a covering map .

###### Proof.

The group contains and is normal in . Thus it must contain by definition. ∎

###### Lemma 4.4.

If a regular tessellation is a link complement, then it is also a universal regular tessellation .

###### Proof.

A regular tessellation is universal if and only if the fundamental group is generated by parabolic elements only. Note that the Wirtinger representation of the fundamental group of hyperbolic link complement implies that it also has this property. ∎

## 5. Classification of Regular Tessellations with Small Cusp Modulus

Once we have constructed a finite-volume universal regular tessellation for a cusp modulus , we can construct all regular tessellation of cusp modulus using the techniques described later in Section 7.2. This is done in Table 2 by listing the following categories to capture the relationships between these regular tessellations:

###### Definition 5.1.

Let denote the category of all pointed manifold regular tessellations of type and cusp modulus . Recall that each regular tessellation is a cover of the orbifold and pick a generic point in the orbifold (that is not on the singular locus or preserved under any symmetries of the orbifold). An object is a regular tessellation of type and cusp modulus such that is a lift of . A morphism is a covering map respecting the base point .

For the categories and , we can describe three objects as quotients by an arithmetically defined group if such a quotient happens to be a manifold. These objects are congruence manifolds with discriminant (for ) or (for ) and among them are the principal congruence manifolds :

 WDz=H3/XDz where WDz=ker(PSL(2,OD)→SL(2,OD/⟨z⟩)/±1) XDz=H3/XDz where XDz=ker(PSL(2,OD)→PSL(2,OD/⟨z⟩)), YDz=H3/YDz where YDz=ker(PGL(2,OD)→PGL(2,OD/⟨z⟩)).

Here, and . For , are the only two elements with , so the map to is well-defined.

###### Lemma 5.2.

If a space respectively, is a manifold, then it is a regular tessellation. Thus, there are covering maps (some might be isomorphisms)

 U{p,q,r}z→WDz→XDz→YDz.
###### Proof.

We need to show that all the groups are normal in the respective . This is obvious for the kernel since and It is also obvious that and are normal in which is an index-2 subgroup of with an element in the complement being . Hence, it is enough to show that the conjugate of a matrix in or is also in , respectively, . This follows from still being in and its image in , respectively, still being congruent to the identity matrix. ∎

If there are other objects in not listed in the above lemma, we denote them by and . The latter one is used for the mirror image of when is chiral. denotes the quaternion group of order 8.

We can now classify all principal congruence link complements as a corollary of the main theorem. It is left to run the algorithm described later in Section 7.1.2 to check whether the arithmetically defined space is an orbifold or manifold. We obtain an orbifold for the arithmetically defined spaces in exactly the following cases: with and .

###### Corollary 5.3.

A principal congruence manifold with and in canonical form is a link complement if and only if:

• and

• and

###### Proof.

By Lemma 5.2 and Theorem 2.4, can only be a link complement if it is a finite volume universal regular tessellation. There are 16 potential cases of type and . In all these cases, is a manifold. But in the case and , it is not the universal regular tessellation, leaving cases for each discriminant. An overview is also given in Figure 4. ∎

## 6. Construction of the Universal Regular Tessellation

In this section, we give an algorithm to construct which will terminate if and only if is finite-volume. Recall the definition of the torus from Definition 3.1.

###### Definition 6.1.

A nanotube is the product where is tessellated by regular -gons when constructing tessellations of type and cusp modulus .

Figure 5 shows an example nanotube. Whereas a regular tessellation is built from ideal Platonic solids, its dual is built from topological nanotubes such that nanotubes meet at each edge and is the cusp modulus. The reader is probably already familiar with the decomposition of a regularly tessellated cusped hyperbolic manifold into nanotubes because the nanotubes happen to be the Ford domains of the manifold. This is analogous to the regular tessellation by Platonic solids being identical to the canonical cell decomposition [EP88]. However, we prefer the dual nanotubes as building blocks in this section because they already have the cusp modulus encoded in them, thus making it easier to build tessellation with prescribed cusp modulus. The use of the dual as well as the term “nanotube” was first suggested by Ian Agol.

To obtain the universal regular tessellation , we can (roughly speaking) just glue enough nanotubes together and enforce that there are always nanotubes at an edge. A deterministic algorithm for this is described in the following definition.

###### Definition 6.2.

Let be a single nanotube. To obtain from , perform the following steps:

1. Attach a nanotube to each open face of .

2. If there is an edge of the resulting complex with nanotubes around and two open faces adjacent to , we need to glue the two faces so we a get an edge cycle about . If we already had an edge cycle about an edge of length different from or there are more than nanotubes adjacent to , we need to identify every -th nanotube. Repeat until there is no edge left for which such a gluing or identification is necessary.

###### Remark 6.3.

is well-defined as the result is independent of the order in which we process the edges in step 2. To see this, note that the space is the quotient of the complex obtained from step 1 under a certain equivalence relationship. Namely, this relationship is the minimal equivalence relationship closed under the operation of rotating a point near any edge to the next nanotube times. It does not matter in which order we perform this operation to obtain the minimal relationship closed under this operation.

The open faces of form a 2-complex, but since we never glue two nanotubes along edges, only along faces, they even form a surface (or potentially, a 2-orbifold, but only for small , see Remark 6.6). This surface is orientable, connected, and tessellated by -gons. We will label an edge of with a numeral 1, …, indicating the number of nanotubes in that are adjacent to .

An example for is shown in Figure 6. In later iterations, there are edges labeled by . Choose one, say . We again attach two nanotubes to the two open faces adjacent to , so we have four nanotubes around now. Thus, we need to identify the two newly attached nanotubes with each other. As before, the edge will become a 3-cycle and disappear from . Effectively, we added only one nanotube glued to the two open faces of adjacent to to close up the edge to a 3-cycle.

###### Remark 6.4.

If the map is an embedding, the edge-labeled tessellation of can be determined purely from , so it is enough to look at the surface for studying the evolution of the algorithm.

###### Lemma 6.5.

Let be a regular tessellation of a finite-volume hyperbolic 3-manifold of type with cusp modulus . There is a map which is unique once we identify with a Ford domain of . For large enough, will be surjective. If is furthermore a universal regular tessellation , then for a large enough . In other words, if is finite volume, there is an such that is empty.

###### Proof.

The existence of is trivial for and follows inductively for from the construction which enforces only the cusp modulus and edge cycle as relations. We say that two cusps of are neighbors if they span an edge in the regular tessellation of . Note that the covers one cusp of , covers that cusp and its neighbors, covers that cusp, its neighbors, and its neighbors’ neighbors and so on. Hence, because is connected and has only finitely many cusps, will eventually be surjective. Now consider the case that is a finite volume universal regular tessellation. Since the only two relations used in the construction of are the cusp modulus and the edge cycle, the map will be an isomorphism for large enough. Since , there will be no open faces in and will be empty. ∎

###### Remark 6.6.

The algorithm can produce an orbifold for small , for example, . consists of a single hexagon. Step 1 doubles the space along the hexagon, thus all edges are closed up and there are two nanotubes about an edge. Step 2 now identifies every third nanotube about an edge and since , there will be only one nanotube adjacent to after identification. Thus, we have introduced singular locus of order 3. There is always a map , but in this case, the map will not be an embedding for . The cases in which is an orbifold are shown in Table 1.

###### Remark 6.7.

If the map is an embedding for all , the construction also gives an algorithm to solve the word problem for the group .

## 7. Computer Implementation

The algorithm described in the previous section has been implemented. The source code and all other files necessary for the reader to easily certify the correctness of the results in this paper are available at http://www.unhyperbolic.org/regTess/. We encourage the reader to read and experiment with the well-commented code for details. Each subsection starts with an example how to use the software followed by implementation details.

Furthermore, we want to point out that regularTessellations.py contains a new triangulation data structure that allows not just gluing tetrahedra but also identifying them (see Remark 7.1). This seems to be a useful feature in general, e.g., for constructing quotient spaces but neither SnapPy [CDW] nor Regina [Bur09] implement it.

### 7.1. Generating Triangulations with regularTessellations.py

#### The Universal Regular Tessellation U{p,q,r}z

Definition 6.2 described the algorithm in terms of nanotubes, but we use triangulations here as they are easier to work with and can also be exported into existing 3-manifold software such as Regina or SnapPy. We obtain the triangulation of a nanotube through the barycentric subdivision, i.e., the subdivision on induced from the barycentric subdivision of . After gluing such subdivided nanotubes, the resulting triangulation is also the barycentric subdivision of the regular tessellation (recall that the unsubdivided nanotubes corresponded to the dual Ford domains but barycentric subdivision is invariant under duality). Note that such a triangulation has finite vertices but still can be imported into SnapPy, as SnapPy performs algorithms to remove finite vertices without changing the topology of the manifold upon import.

The following example shows how this triangulation of the universal regular tessellation, here , can be constructed. As regular tessellation, consists of 28 regular ideal tetrahedra. Thus its barycentric subdivision consists of 672 simplices, each ideal tetrahedron contributing 24. The last two lines of code convert the data to a Regina triangulation and write it to a file that can be read with SnapPy. Only those last two lines actually depend on Regina being installed, whereas all other methods work in pure Python 2.x:

>>> from regularTessellations import *
>>> tessConext = TessellationContext(3,3,6,2,1)
>>> tess = tessContext.UniversalRegularTessellation()
>>> len(tess), len(tess)/24
672, 28
>>> reginaTrig = TetrahedraToReginaTriangulation(tess)
>>> open(’2_plus_1_zeta_tets.trig’,’w’).write(reginaTrig.snapPea())


We now describe in more detail how this is implemented and refer the reader to the python code for details. We first need to write a nanotube factory that produces the triangulation of a nanotube obtained by barycentric subdivision. An algorithm to create the resulting triangulation is easily implemented and we spare the reader with the details, only mentioning the conventions we use here (also see Figure 3): We label the vertices of a simplex in this subdivision such that vertex 0 is ideal, 1 at the center of a face of the nanotube, 2 at an edge adjacent to the face, and 3 at a vertex adjacent to the edge. Thus, face opposite to vertex is always glued to face of another simplex such that the face pairing permutation is trivial, i.e., such that the vertex is glued to vertex , and the only gluing data we need to store per simplex is one reference to another simplex per face. We also label the simplices with an orientation so that two neighboring simplices always have opposite orientations since each face gluing reverses the orientation. Every face 0 of a newly created nanotube is unglued.

Step 1 of Definition 6.2 just invokes the nanotube factory to create a new nanotube for each open -gon of and then glues some -gon of to . An -gon is formed by simplices, so this involves gluing simplices of to simplices of along face 0 and we must be careful to glue them in such a way that a positively oriented simplex is glued to a negatively oriented simplex.

Then we need to apply step 2 to each simplex . Let be the simplex glued to face of . If the face is unglued, we say that does not exist. Similarly, let be the simplex glued to face of simplex . If simplex exists, then identify with if they are not already identified. Otherwise, glue to along face 0 if face 0 of is unglued and exists. Here, strings such as “1010…1” are supposed to contain “1” times.

We need to iterate step 2 until no identification or gluing happened in an iteration. This is because gluing up one edge might trigger that nearby edges have more adjacent simplices and need to be glued up.

It is left to write a loop that repeats step 1 and 2 until there are no open faces.

###### Remark 7.1.

When we identify two simplices and , the identification needs to be pushed through the already existing gluings, e.g., if is glued to along face 0 and is glued to , then and need to be identified as well if they are different, and this identification then needs to be recursively pushed through as well. If only is glued to a simplex , but face 0 of is unglued, the simplex resulting from identifying and will be glued to .

#### Arithmetically Defined Regular Tessellations

The following example shows how to construct and . We see that is a double cover of (they are actually the manifold and orbifold in Figure 1).

>>> tessContext = TessellationContext(3,3,6,2,0)
>>> X = tessContext.PrincipalCongruenceManifold(’X’)
>>> len(X)
240
>>> tessContext = TessellationContext(3,3,6,2,0)
>>> Y = tessContext.PrincipalCongruenceManifold(’Y’)
>>> len(Y)
120


The algorithm works as follows: To construct respectively, , take the triangulation of from Section 7.1.1 with large enough and fix a positively oriented simplex of it, say . Lift to the universal cover such that it becomes the simplex in Figure 3 (for , vertex 3 is above instead of ). We can now label each simplex with positive orientation by a matrix that would take to in . To start, label by the identity. Let

 gP=(01−11),gQ=(−1/u101)andgR=(u001).

They correspond to the rotations indicated in Figure 3. Assume is labeled by . Let be the simplex glued to face of and be the simplex glued to face of . We assign the label to , to , to , to , to and to . We need to identify two simplices if the images of their labels in differ by a matrix in or . We can represent the image in by a matrix such that the determinant is either 1 or . Replace each label by such a matrix.

It turns out that the coefficients of the labels explode when computing the products. Hence, we instead label the simplices by pairs where is normalized as above such that is either 1 or . We need to store the determinant here as well, because it is not determined by alone for small . Now identify two simplices if their labels have

• the same up to and the same (for )

• the same up to and the same (for )

• the same up to (for ).

If there are open faces left, was not chosen large enough.

#### Orbifold Detection

For a triangulation returned by one of the above algorithms, we can check the number of simplices around each edge. If these numbers are for the respective edges of each simplex, it is the triangulation of a hyperbolic manifold, otherwise, an orbifold. This can be checked as follows (continuing previous example, recall that was the orbifold and the manifold in Figure 1):

>>> tessContext.IsManifold(Y)
False
>>> tessContext.IsManifold(X)
True


### 7.2. Finding All Regular Tessellations Using regularTessellations.g

Recall that every regular tessellation of type and cusp modulus can be obtained as quotient of by a normal subgroup where is the orientation-preserving symmetry group of . If is finite-volume, we can use Gap [GAP08] to find all suitable normal subgroups and thus classify all regular tessellations in these cases. Here is an example for and :

gap> Read("regularTessellations.g");
gap> G:=SymmetriesUniversalRegularTessellationPermGroup(3,3,6,4,0);
<permutation group of size 7680 with 3 generators>
gap> L:=AllRegularTessellationsFromUniversalRegularTessellation(G);;
gap> List(L,StructureDescription);
[ "1", "C2", "C4" ]
gap> IsSubgroup(L[3],L[2]);
true


This shows that there are three manifold regular tessellations with cusp modulus : and two extra manifolds covered by such that the Decktransformations are , respectively, . We also see that one of these extra manifolds is covered by the other one.

Here, we use the following representation for (where ):

 (1) G≅⟨P,Q,R|Pp,Qq,Rr,(PQ)2,(QR)2,(PQR)2,(QRr/2+1)a(RQRr/2)b⟩.

Fixing a simplex of , the three generators can be identified with rotations about edges of as shown in Figure 3. Namely, a right multiplication by means we go along face 0 and 1. Analogously, goes along face 1 and 2, and along face 2 and 3.

Given a normal subgroup of , we need to check that the quotient is a manifold regular tessellation of the same cusp modulus. Let , , , , and be the cyclic subgroups of the rotations about one of the six edges of . Let be the subgroup of elements fixing the cusp corresponding to vertex 0 of . Then, the quotient by a normal subgroup is a manifold if and only if intersects each of the six trivially and has the same cusp modulus if and only if intersects trivially.

It should be noted that a quotient can be chiral even though is amphicheiral. Here is an example for regular tessellations of type and cusp modulus :

gap> G:=SymmetriesUniversalRegularTessellationPermGroup(4,3,6,2,0);;
gap> L:=AllRegularTessellationsFromUniversalRegularTessellation(G);;
gap> IsAmphicheiralRegularTessellation(L[4],G);
false
gap> mu:=MirrorIsomorphismUniversalRegularTessellation(G);;
gap> Image(mu,L[4])=L[5];
true


We see that the fourth regular tessellation in the list is chiral and that its mirror image is the fifth regular tessellation in the list (Table 2 lists them as and ).

To detect this in general when is amphicheiral, let be the group automorphism obtained by conjugating with an orientation-reversing symmetry:

 μ:G→G,P↦Q−1P−1Q,Q↦Q−1,R↦R−1.

Now the mirror image of a quotient is given by and the quotient is amphicheiral if and only if .