# Exact enumeration of fullerenes

Philip Engel  University of Georgia<EMAIL_ADDRESS>and  Peter Smillie
Heidelberg University<EMAIL_ADDRESS>KU Leuven
<EMAIL_ADDRESS>
###### Abstract.

A fullerene, or buckyball, is a trivalent graph on the sphere with only
pentagonal and hexagonal faces. Building on ideas of Thurston, we use modular
forms to give an exact formula for the number of oriented fullerenes with a
given number of vertices.

##  1\. Introduction

We call a triangulation $\mathcal{T}$ of $S^{2}$ convex if every vertex has
valence $6$ or less. The curvature of a vertex $v_{i}\in\mathcal{T}$ is
$\kappa_{i}:=6-{\rm valence}(v_{i})$ and the curvature profile
$\\{\kappa_{1},\dots,\kappa_{m}\\}$ is the multiset of nonzero curvatures. It
is a consequence of Euler’s formula that $\sum\kappa_{i}=12$ and so there are
at most $12$ vertices of positive curvature. We call the dual complex of a
triangulation with exactly $12$ vertices of positive curvature a buckyball or
fullerene—a trivalent graph on the sphere, whose faces are all pentagons and
hexagons.

The terminology originates from the famous Buckminsterfullerene molecule, with
chemical formula ${\rm C}_{60}$. It is formed from $60$ carbon atoms, each one
bonded to three other carbon atoms in $sp^{2}$-hybridized orbitals, and
forming a spherical molecule with carbon rings of length $5$ and $6$, see
Figure 1. The molecule was first discovered in the laboratory in 1985 by
Kroto, Heath, O’Brien, Curl, and Smalley [13], work which received the 1996
Nobel prize in chemistry.

Figure 1. The Buckminsterfullerene molecule

###  1.1. Algorithmic enumeration

Soon after the discovery of fullerenes, the question of their enumeration was
explored by chemists, computer scientists, and mathematicians, resulting in a
large body of literature: see Liu, Klein, Schmalz, and Seitz [15],
Manolopoulos and May [19], Manolopoulos and Fowler [17, 18], Sah [20], Yoshida
and Osawa [28], Brinkmann and Dress [3], Hasheminezhad, Fleischner, and McKay
[12], Brinkmann, Goedgebeur, and McKay [4, 11].

The earliest algorithms, proposed by Manolopoulos et al. enumerated fullerenes
by spiraling outward from a face, as one might peel an orange. A spiral
representation $[i_{1},\dots,i_{12}]$ of the fullerene is then simply the list
of $12$ indices corresponding to positions of the 12 pentagons in a face
spiral of the given fullerene. The canonical spiral representation is the
lexicographically minimal spiral representation. For example, the canonical
spiral representation of Buckminsterfullerene is

$[1,7,9,11,13,15,18,20,22,24,26,32].$

Remarkably, there are fullerenes with no spiral representation, the first
counterexample being an isomer of ${\rm C}_{380}$ with tetrahedral symmetry
[5].

The spiral algorithm was supplanted by Brinkmann and Dress’s algorithm,
implemented in the program fullgen. It generates all fullerenes with a given
number of vertices by stitching together patches which are bounded by zigzag
(Petrie) paths. This was again supplanted by buckygen, developed by Brinkmann,
Goedgebeur, and McKay [4]. It generates larger fullerenes from smaller ones,
by excising a patch of faces and replacing it with a larger patch having the
same boundary. There are three irreducible fullerenes in the buckygen
algorithm, isomers of ${\rm C}_{20}$, ${\rm C}_{28}$, and ${\rm C}_{30}$.

Because of its recursion, buckygen is much more efficient than fullgen,
especially for generating fullerenes up to $n$ vertices. Contradicting results
of buckygen and fullgen led to the detection of a small programming error in
fullgen, which led to a discrepancy in the counts beyond $136$ vertices. After
this was fixed, the two programs agreed up to 380 vertices, giving a high
degree of confidence in their accuracy.

The input to this paper was generated with buckygen, see Appendix A by Jan
Goedgebeur, with a built-in setting that counts enantiomorphic (mirror)
fullerenes as distinct. This makes the mathematical enumeration somewhat less
complicated; we hope to return to the case of enumerating unoriented
fullerenes in future work.

###  1.2. Mathematical enumeration

Thurston’s paper “Shapes of Polyhedra” [27] introduced the moduli spaces
$\mathcal{M}_{\alpha_{1}\,,\,\dots\,,\,\alpha_{k}}$ of flat cone spheres
(modulo scaling): These are flat metrics on $S^{2}$ with $k$ conical
singularities $p_{i}$ of cone angles $2\pi-\alpha_{i}$ and
$\alpha_{i}\in(0,2\pi)$. The Gauss-Bonnet formula implies
$\sum\alpha_{i}=4\pi$. Alexandrov’s theorem [1] states that a flat cone sphere
is isometric to a convex polyhedron in $\mathbb{R}^{3}$, unique up to rigid
motion.

Thurston proved that $\mathcal{M}_{\alpha_{1}\,,\,\dots\,,\,\alpha_{k}}$
admits a canonical Hermitian metric, endowing it with the structure of a
locally symmetric space: It admits a local isometry [27, Thm. 0.2] to complex
hyperbolic space

$\mathbb{CH}^{k-3}:=\mathbb{P}\\{v\in\mathbb{C}^{1,{k-2}}\,\big{|}\,v\cdot
v>0\\},$

also called the complex ball. The metric completion
$\overline{\mathcal{M}}_{\alpha_{1}\,,\,\dots\,,\,\alpha_{k}}$ is a complex
hyperbolic cone manifold, and admits a moduli-theoretic interpretation,
corresponding to allowing collections of cone points $\\{p_{i}\,\big{|}\,i\in
I\\}$ to coalesce, so long as $\sum_{i\in I}\alpha_{i}<2\pi$.

The completed Thurston moduli space of interest to this paper is

$M:=\overline{\mathcal{M}}_{\pi/3\,,\,\dots\,,\,\pi/3}$

as every convex triangulation defines a point in it, by declaring each
triangle metrically equilateral of side length $1$. A convex triangulation
corresponds to a fullerene if and only if the corresponding flat cone sphere
has $12$ distinct points of positive curvature, or equivalently, defines a
point in the open stratum $\mathcal{M}_{\pi/3\,,\,\dots\,,\,\pi/3}$. There are
$48$ possible curvature profiles for convex triangulations of the two sphere
$S^{2}$, in bijection with the $48$ partitions
$\\{\kappa_{1},\dots,\kappa_{m}\\}$ of $12$ into parts of size $5$ or less.
Each such partition corresponds to a curvature stratum:

$M=\overline{\mathcal{M}}_{\pi/3\,,\,\dots\,,\,\pi/3}=\\!\\!\\!\\!\\!\\!\coprod_{\begin{subarray}{c}\sum\kappa_{i}=12\\\
\kappa_{i}\in\\{1,2,3,4,5\\}\end{subarray}}\\!\\!\\!\\!\\!\\!\mathcal{M}_{\kappa_{1}\pi/3\,,\,\dots\,,\,\kappa_{m}\pi/3}.$

Thurston proved that $M=\mathbb{P}\Gamma\backslash\mathbb{CH}^{9}$ is a global
ball quotient for an arithmetic subgroup $\Gamma\subset U(1,9)$; it is the
largest-dimensional completed Thurston moduli space which is a global
quotient. See also [22].

######  Definition 1.1.

An Eisenstein lattice $\Lambda$ is a finitely generated
$\mathbb{Z}[\zeta_{6}]$-module $\Lambda\simeq\mathbb{Z}[\zeta_{6}]^{k}$,
together with a non-degenerate $\mathbb{Z}[\zeta_{6}]$-valued Hermitian inner
product $v\star w$. Its signature $(m,n)$ is the signature of
$\Lambda\otimes_{\mathbb{Z}[\zeta_{6}]}\mathbb{C}\simeq\mathbb{C}^{m,n}$.

######  Theorem 1.2 ([27, Thm. 0.1]).

There is an Eisenstein lattice $\Lambda$ of signature $(1,9)$ for which convex
triangulations of the sphere, up to oriented isomorphism, are in bijection
with $\Gamma\backslash\Lambda^{+}$. Here $\Lambda^{+}\subset\Lambda$ is the
set of vectors of positive norm and $\Gamma=U(\Lambda)$ is the group of
unitary isometries.

Furthermore, for a vector $v\in\Lambda^{+}$ corresponding to a triangulation,
the number of triangles is $\tfrac{2}{3}(v\star v)$.

The point in $\mathbb{P}\Gamma\backslash\mathbb{CH}^{9}=M$ corresponding to
the triangulation is the $\Gamma$-orbit of the complex line
$\mathbb{C}v\in\mathbb{CH}^{9}$.

Thurston’s theorem bounds the growth rate for the number of fullerenes with
$2n$ vertices as $O(n^{9})$. The computational chemists were aware of this
fact and Thurston’s result, see for instance the survey of Schwerdtfeger,
Wirz, and Avery [23].

In 2017, the authors used Thurston’s theorem to enumerate convex
triangulations, when counted with an appropriate weight:

######  Definition 1.3.

Define the mass of a convex triangulation to be

$m(\mathcal{T}):=\frac{1}{|{\rm
Aut}^{+}(\mathcal{T})|}\prod_{i=1}^{m}\frac{(1-\kappa_{i}/6)^{\kappa_{i}-1}}{\kappa_{i}!}$

where $\kappa_{i}$ are the curvatures and ${\rm Aut}^{+}(\mathcal{T})$ is the
group of oriented automorphisms. Alternatively, we can define
$m(\mathcal{T})={|{\rm Stab}_{\Gamma}(v)|}^{-1}$ where $v\in\Lambda^{+}$
corresponds to $\mathcal{T}$.

######  Theorem 1.4 ([10, Thm. 1.1]).

The mass-weighted sum of all convex triangulations with $2n$ triangles is
exactly

$\sum_{|\mathcal{T}|=2n}m(\mathcal{T})=\frac{809}{2^{15}3^{13}5^{2}}\sum_{d\mid
n}d^{9}.$

The key point of the proof is to use the Siegel theta correspondence to show
that the mass-weighted generating function $f(q)$ for convex triangulations is
a modular form of weight $10$ for $SL_{2}(\mathbb{Z})$. Up to a constant
scaling factor, this pins down $F(q)$ uniquely as the Eisenstein series
$E_{10}(q)=-\frac{1}{264}+\sum_{n\geq 1}\big{(}\sum_{d\mid
n}d^{9}\big{)}q^{n}$. To determine the constant, it suffices to compute the
the weighted count of triangulations with $2$ triangles.

Let $G$ be a group acting on a set $S$, with finitely many orbits $\\{G\cdot
s\,\big{|}\,s\in S\\}$, and finite stabilizers. It is most natural to weight
the count of orbits by $|{\rm Stab}_{G}(s)|^{-1}$ where $s\in S$ is an orbit
representative. This way, passing to a finite index subgroup $H\subset G$
changes the weighted count by the index $[G:H]$. It is this manner of
enumeration employed in Theorem 1.4, with $G=U(\Lambda)$ and $S=\Lambda^{+}$.

###  1.3. Main results

The enumeration of fullerenes by chemists is not a mass-weighted count. In
contrast to the mass-weighted enumeration, we call an unweighted count of
orbits naive enumeration.

######  Definition 1.5.

Let ${\rm BB}(q):=\sum_{n\geq 1}{\rm BB}_{n}q^{n}$ where ${\rm BB}_{n}$ is the
naive count of oriented fullerenes with $2n$ vertices. We have

${\rm
BB}(q)=q^{10}+q^{12}+q^{13}+3q^{14}+3q^{15}+10q^{16}+9q^{17}+23q^{18}+\cdots.$

Obviously, ${\rm BB}(q)\in\mathbb{Z}[[q]]$ has integer coefficients, while the
weighted generating function for convex triangulations
$F(q)\in\mathbb{Q}[[q]]$ does not. Even though the $q^{n}$-coefficients of
$F(q)$ and ${\rm BB}(q)$ as asymptotic to each other as $n\to\infty$, the
series differ in two key ways:

  1. (1)

Counts for $F(q)$ are over the completed space
$M=\overline{\mathcal{M}}_{\pi/3\,,\,\dots\,,\,\pi/3}$ whereas counts for
${\rm BB}(q)$ are over the open stratum
$\mathcal{M}_{\pi/3\,,\,\dots\,,\,\pi/3}$.

  2. (2)

Counts for $F(q)$ are weighted, but counts for ${\rm BB}(q)$ are naive. Both
$\\{\kappa_{1},\dots,\kappa_{m}\\}$ and ${\rm Aut}^{+}(\mathcal{T})$ affect
the weight.

Our goal is to surmount these obstacles, and count fullerenes naively. While
conceptually simpler, this is mathematically much more complicated.

The main idea is to prove that for each curvature profile and automorphism
group, there is an appropriately weighted count of such triangulations which
is again a (quasi)modular form, but of lower weight and higher level. Then, by
Möbius inversion on strata, the naive count ${\rm BB}(q)$ has an expression as
a weighted inclusion-exclusion of weighted counts. So ${\rm BB}(q)$ is a
linear combination of modular forms, but of mixed levels and weights. Sah [21,
Thm. 3.4] was the first to realize ${\rm BB}_{n}$ would have a formula of an
arithmetic nature.

Our formula is proven by bounding the levels and weights of these modular
forms explicitly, and using the buckygen program as input—we solve a large
linear system for the coefficients of the linear combination. Fortuitously,
buckygen can compute $200$ Fourier coefficients ${\rm BB}_{n}$. This exceeds
the $152$-dimensional space of modular forms in which we prove ${\rm BB}(q)$
lives. Once a formula in a finite-dimensional space is established, the
ambient space of modular forms is pruned, resulting in our main theorem:

######  Theorem 1.6.

${\rm BB}(q)$ is an explicit linear combination of Eisenstein series, of
weight at most $10$ and levels lying in $\\{1,2,3,4,5,6,9,12,15,18\\}.$

More precise statements are given in the body of the paper, see Theorem 3.25.
Using this explicit formula, we can compute the number of oriented fullerenes
far beyond buckygen. In fact, evaluating the formula is computationally only
as hard as factoring $n$.

##  2\. Strata of triangulations

We must understand the stratification of
$M=\overline{\mathcal{M}}_{\pi/3\,,\,\dots\,,\,\pi/3}=\mathbb{P}\Gamma\backslash\mathbb{CH}^{9}$
by curvatures and automorphism groups in more detail.

######  Definition 2.1.

The automorphism stratification of $M$ is the stratification by oriented
automorphism groups ${\rm Aut}^{+}(\mathcal{T})$. Let $M_{G}\subset M$ denote
the stratum of flat cone spheres with ${\rm Aut}^{+}(\mathcal{T})=G$. So
$M=\coprod_{G}M_{G}$.

A flat cone sphere can be specified by a Lauricella differential, i.e. a
multivalued differential $\omega$ on $\mathbb{P}^{1}$ with fractional pole
orders (see [7, 16]). Such a differential defines a flat structure
$\mathcal{T}$ on $\mathbb{P}^{1}\simeq S^{2}$ by declaring $z$ a local flat
coordinate if $dz=\omega$, i.e. the flat metric is $|\omega|^{2}$. Two
Lauricella differentials $\omega_{1}$ and $\omega_{2}$ determine the same flat
cone sphere, mod scaling, if and only if $\omega_{1}=c\omega_{2}$ for
$c\in\mathbb{C}^{*}$. Conversely, every flat cone sphere is naturally a
Riemann surface isomorphic to $\mathbb{P}^{1}$, and defines a Lauricella
differential by declaring $\omega=dz$ for a local flat coordinate $z$.

Given any 12 points $p_{i}\in\mathbb{P}^{1}$, no more than 5 of which are
simultaneously equal, and any constant $c\in\mathbb{C}^{*}$, we form the
Lauricella differential

(1) $\omega=c(x-p_{1})^{-1/6}\cdots(x-p_{k})^{-1/6}\,dx$

in which the product is over $p_{i}\neq\infty$. Then the resulting flat cone
sphere lies in $M$, and conversely every point of $M$ can be described by a
Lauricella differential of this form with $c=1$. Thus, $M$ also admits a
description as the space of $12$ unordered points on $\mathbb{P}^{1}$, in
which up to $5$ points can coalesce, up to the action of the holomorphic
automorphism group of $\mathbb{P}^{1}$.

Note that such a differential $\omega$ is multivalued and that
$\omega=\zeta\omega$ for $\zeta$ any sixth root of unity.

######  Proposition 2.2.

The possible automorphism groups ${\rm Aut}^{+}(\mathcal{T})$ for some
$\mathcal{T}\in M$ are precisely:

  1. (1)

a cyclic group $C_{n}$ for $n\in\\{1,2,3,4,5,6,7,8,9,10,11\\}$,

  2. (2)

a dihedral group $D_{n}$ for $n\in\\{2,3,4,5,6,8,10,12\\}$, or

  3. (3)

one of the three exceptional groups $T$, $O$, $I$, which are respectively
symmetries of the tetrahedron, octahedron, icosahedron.

###### Proof.

The three exceptional groups are realized by putting $p_{1},\dots,p_{12}$ at
the vertices of the tetrahedron, octahedron, and dodecahedron respectively.
Note that for $T$, three points coincide at each vertex, and for $O$, two
points coincide at each vertex.

The remaining finite subgroups of $PGL_{2}(\mathbb{C})$ are $C_{n}$ and
$D_{n}$. The orbits of the $C_{n}$ action, acting by $x\mapsto\zeta_{n}x$, are
the singleton poles $\\{0\\}$, $\\{\infty\\}$ and free orbits of size $n$.
Since no more than $5$ points can coalesce at $0$ or $\infty$, there must be
at least one free orbit and so $n\leq 12$. If $n=12$, then the points $p_{i}$
lie in a single free orbit of $C_{12}$, in which case $D_{12}$ symmetry is
automatic. For $C_{n}$ with $n\leq 11$, we can always put some points $p_{i}$
at the poles $0$, $\infty$ with non-equal weights less than $6$, and add a
union of free orbits, to get a flat cone sphere $\mathcal{T}\in M$ with cyclic
$C_{n}$ symmetry.

To realize $D_{n}$ symmetry with $n\in\\{2,3,4,5,6\\}$, we place $6-n$ points
at each of $0$, $\infty$ and place the remaining $2n$ points on a free $D_{n}$
orbit. To realize $D_{n}$ symmetry with $n\in\\{8,10,12\\}$, we place
$\frac{12-n}{2}$ points at each of $0$, $\infty$ and put the remaining $n$
points along the equator in a $D_{n}$ orbit of size $n$. The other dihedral
groups $D_{n}$ for $n\in\\{7,9,11,13,14,\dots\\}$ are not realizable because
either they contain $C_{n}$ with $n\geq 13$ or the weights at $0$, $\infty$
cannot be made equal for parity reasons. ∎

Not all automorphism groups of flat cone spheres in $M$ are realized as
automorphism groups of triangulations:

######  Example 2.3.

The cyclic group $C_{11}$ acts on $\mathbb{P}^{1}$ by rotations, with a
generator acting by $x\mapsto\zeta_{11}x$. There is a flat cone sphere with
one point at $\infty$ and $11$ points on the equator, all of curvature $1$,
associated to the Lauricella differential $(1-x^{11})^{-1/6}\,dx$. This flat
cone sphere is not realized by an equilateral triangulation.

If $\omega$ is a Lauricella differential of the form $\eqref{eqn:lauricella}$,
then the integral of $\omega$ along any path is well-defined up to a sixth
root of unity. By the periods of $\omega$, we mean the integrals along paths
between singular points, considered as numbers in
$\mathbb{C}/\langle\zeta_{6}\rangle$. It follows from [27] that:

######  Proposition 2.4.

Equilaterally triangulated flat cone spheres are equivalent to Lauricella
differentials of the form (1), all of whose periods lie in
$\mathbb{Z}[\zeta_{6}]$.

###### Proof.

If $\mathcal{T}$ is triangulated, we choose the Lauricella differential
$\omega$ so that the period map sends any given triangle of $\mathcal{T}$ onto
a triangle of the standard triangulation of $\mathbb{C}$. (Recall that
$\omega$ is a priori determined only up to a complex number of norm one). Then
the developing map sends all singularities of $\mathcal{T}$ onto points of
$\mathbb{Z}[\zeta_{6}]$, i.e. the periods of $\omega$ will lie in
$\mathbb{Z}[\zeta_{6}]$.

In the other direction, if all the periods of $\omega$ lie in
$\mathbb{Z}[\zeta_{6}]$, then we recover the triangulation by declaring a
point $p\in\mathcal{T}$ to be a vertex if the integral of $\omega$ from any
singularity to $p$ lies in $\mathbb{Z}[\zeta_{6}]$, and a path from one vertex
to another to be an edge if the integral of $\omega$ along that path takes
values between 0 and 1, up to the indeterminacy of $\langle\zeta_{6}\rangle$.
∎

######  Proposition 2.5.

Suppose that a flat cone sphere $\mathcal{T}$ admits an equilateral
triangulation. Then ${\rm Aut}^{+}(\mathcal{T})$ preserves the triangulation.

###### Proof.

Let $\omega=c\prod_{i}(x-p_{i})^{-1/6}\,dx$ be the Lauricella differential
associated to the triangulation. Oriented automorphisms of $\mathcal{T}$ are
exactly given by $g\in PGL_{2}(\mathbb{C})$ for which
$g^{*}\omega\in\mathbb{C}\omega$ or equivalently
$g^{*}\\{p_{i}\\}=\\{p_{i}\\}$. Since $\gamma$ has finite order, we
necessarily have $g^{*}\omega=\zeta\omega$ for some root of unity $\zeta$,
well-defined up to a sixth root of unity. When $\mathcal{T}$ is triangulated,
then $g$ preserves the triangulation iff $\zeta=1$ (up to a sixth root of
unity).

Let $\gamma$ be any path between singularities of $\mathcal{T}$ with nonzero
period. Since

$\int_{g^{*}\gamma}\omega=\int_{\gamma}g^{*}\omega=\zeta\int_{\gamma}\omega$

and all periods of $\omega$ are in $\mathbb{Z}[\zeta_{6}]$, we see that
$\zeta\in\mathbb{Q}[\zeta_{6}]$. But the only roots of unity in
$\mathbb{Q}[\zeta_{6}]$ are sixth roots of unity, so $g$ must preserve the
Lauricella differential, and hence the triangulation. ∎

If $\mathcal{T}$ is a flat cone sphere, and
$G\subset\mathrm{Aut}^{+}(\mathcal{T})$, then the quotient can also be given
the structure of a flat cone sphere, where the cone angle at a point in the
quotient is the cone angle upstairs divided by the size of its stabilizer.
However, we will also find it useful to consider the quotient as an orbifold.
To that end, we define:

######  Definition 2.6.

A flat cone orbifold is a flat metric on $S^{2}$ with $k$ conical
singularities of cone angle $2\pi-\alpha_{i}$ and specified “ramification
orders” $n_{i}$ subject to the condition that
$n_{i}(2\pi-\alpha_{i})\in(0,2\pi]$ and $\alpha_{i}>0$.

A local orbifold chart near such a cone point is given by an $n_{i}$-to-$1$
branched cover from a cone of angle $n_{i}(2\pi-\alpha_{i})$, which is a local
isometry away from the branch point.

######  Definition 2.7.

An orbitriangulation of a flat cone orbifold is a system of compatible
equilateral triangulations in every possible orbifold chart.

For the orbifold $\mathcal{T}/G$, an orbitriangulation of $\mathcal{T}/G$ is
the same thing as a $G$-invariant triangulation of $\mathcal{T}$. By
Proposition 2.5, every triangulation of $\mathcal{T}$ is $G$-invariant, so in
particular we have

######  Corollary 2.8.

The quotient of a triangulated flat cone sphere in $\mathcal{T}\in M$ by a
subgroup of its automorphism group ${\rm Aut}^{+}(\mathcal{T})$ is an
orbitriangulated flat cone sphere in $M$.

Figure 2. An orbitriangulation (left) and a triangulation of the underlying
flat cone sphere.

######  Example 2.9.

Figure 2 (left) shows an orbitriangulation. The points are labeled with their
ramification orders $n$ and their curvature $\mu$, which is related to the
cone angle deficit $\alpha$ by $\alpha/(2\pi)=\mu/6$. The unlabeled vertices
are identified with the $(n=1,\mu=2)$ vertex by the gluing, which identifies
the dotted lines in pairs and the solid lines as marked. As required, we have
$n(6-\mu)\leq 6$ at each point.

Since the orbifold indices are $(3,2,2)$, this is equal to
$\mathbb{P}^{1}/D_{3}$ as an orbifold. The corresponding $D_{3}$-invariant
triangulation upstairs has $6$ singular points $p_{i}$ each of curvature
$\kappa_{i}=2$, the preimages of the $(n=1,\mu=2)$ point. In this example,
none of the orbifold points are singular upstairs.

######  Remark 2.10.

A flat cone orbifold is not necessarily the quotient of a manifold (i.e. a
_good orbifold_). For example, in the same picture of Figure 2, if we had
declared the vertex between the two arrows to instead be $(n=1,\mu=3)$, this
would be an orbitriangulated flat cone orbifold that is not good. By the
convexity inequality, every good flat cone orbifold is the quotient either of
the plane (if $n_{i}(2\pi-\alpha_{i})=2\pi$ for all $i$) or of a flat cone
sphere.

Note that an orbitriangulated flat cone sphere also admits a triangulation,
but by triangles of side length $\tfrac{1}{2\sqrt{3}}$ rather than $1$. See
Figure 2 (right).

######  Remark 2.11.

As we can see from Example 2.3, if $\mathcal{T}$ is not triangulated, then its
quotient by its automorphism group may not even lie in $M$. For example, the
quotient by the $C_{11}$ action on the flat cone sphere
$(1-x^{11})^{-1/6}\,dx$ lies in
$\mathcal{M}_{61\pi/33\,,\,20\pi/11\,,\,\pi/3}$.

######  Proposition 2.12.

The poset diagram of automorphism groups of convex triangulations is:

${C_{1}}$${C_{2}}$${C_{3}}$${D_{2}}$${D_{3}}$${D_{5}}$${D_{4}}$${T}$${D_{6}}$${O}$${I}$

###### Proof.

We first observe that any element $g\in{\rm Aut}^{+}(\mathcal{T})$ has order
at most $6$. This is because any $g\in PGL_{2}(\mathbb{C})$ fixes some point
$p$. Assuming $g\neq 1$, either $p$ is the center of a triangular face and
${\rm ord}(g)=3$, or $p$ is the center of an edge and ${\rm ord}(g)=2$, or $p$
is a vertex and ${\rm ord}(g)$ divides ${\rm val}(p)\leq 6.$

Conversely, we can easily construct convex triangulations on which $C_{n}$ and
$D_{n}$ act for $n\leq 6$, for instance $n$-gonal bipyramids, and there are
triangulated platonic solids with automorphism groups $T$, $O$, $I$.

But, note that $C_{4}$, $C_{5}$, $C_{6}$ do not appear in the diagram. The
reason is the following: Any cyclic symmetry of order $4$, $5$, $6$ must fix a
vertex of the triangulation of valence $4$, $5$, $6$, at both of the poles of
the cyclic rotation. There are either one or two remaining free $C_{4}$,
$C_{5}$, $C_{6}$ orbits. Thus $C_{4}$, $C_{5}$, $C_{6}$ symmetry always
implies $D_{4}$, $D_{5}$, $D_{6}$ symmetry—there is always an involution
switching the two poles and the remaining free orbit(s). ∎

We now analyze how automorphism groups and curvatures interact.

######  Definition 2.13.

The bistratification of $M=\mathbb{P}\Gamma\backslash\mathbb{CH}^{9}$ is the
stratification by conjugacy classes of inertia groups ${\rm
Stab}_{\mathbb{P}\Gamma}(p)$ where $p\in\mathbb{CH}^{9}$.

######  Proposition 2.14.

The bistrata are the intersections of automorphism strata and curvature
strata.

###### Proof.

The local stabilizer of some $[\mathcal{T}]\in M$ corresponding to
$\mathbb{C}v\in\mathbb{CH}^{9}$ has a canonical extension structure

$0\to\textstyle\prod_{i}B_{\kappa_{i}}\to{\rm
Stab}_{\mathbb{P}\Gamma}(\mathbb{C}v)\to{\rm Aut}^{+}(\mathcal{T})\to 0$

where $B_{\kappa_{i}}$ is the image in $U(\Lambda)$ of the local braid group
of the $\kappa_{i}$ colliding singularities. Thus, ${\rm
Stab}_{\mathbb{P}\Gamma}(p)$ jumps in size exactly when (1) singularities
further collide or (2) the size of the automorphism group increases.∎

A bistratum of $M$ is specified by the list of pairs $\\{(n_{i},\mu_{i})\\}$
of ramification orders and curvatures, at each of the singularities of the
quotient flat cone orbifold. Such a list determines a nonempty bistratum if
and only if

  1. (1)

The multiset of ramification orders $\\{n_{i}\\}$ not equal to 1 are those of
a good orbifold $\mathbb{P}^{1}/G$, for one of the groups $G$ in Proposition
2.12,

  2. (2)

for each $i$, $0<\mu_{i}<6$ and $n_{i}(6-\mu_{i})\leq 6$, and

  3. (3)

$\sum_{i}\mu_{i}=12$.

The automorphism group $G$ of the corresponding bistratum is determined by the
$\\{n_{i}\\}$. The curvature profile $\\{\kappa_{j}\\}$ of the flat cone
sphere upstairs is determined as follows: Each singular point of the quotient
orbifold with indices $(n_{i},\mu_{i})$ satisfying $n_{i}(6-\mu_{i})<6$
corresponds to $|G|/n_{i}$ singular points upstairs, each with curvature
$\kappa_{j}$ given by the formula

$6-\kappa_{j}=n_{i}(6-\mu_{i}).$

The singular points of the quotient orbifold with $n_{i}(6-\mu_{i})=6$ are
non-singular upstairs.

######  Notation 2.15.

We introduce an alternative notation for the bistratum
$\\{(n_{i},\mu_{i})\\}$. First, instead of listing the ramification orders
$\\{n_{i}\\}$, we write the corresponding group $G$. Second, we order the
$\mu_{i}$ so that they correspond to the ramification orders of
$\mathbb{P}^{1}/G$ in descending order. Third, we add a superscript $\bf e$ to
the index $\mu_{i}=3$ whenever $n_{i}=2$, and we add a superscript $\bf f$ to
the index $\mu_{i}=4$ whenever $n_{i}=3$.

So, for example, the bistratum $\\{(3,4),(2,3),(2,3),(1,2)\\}$ of the orbifold
depicted in Figure 2 is written $(D_{3},\\{4^{\bf f},3^{\bf e},3^{\bf
e},2\\})$. We will call the list $\\{\mu_{i}\\}$ with superscripts a
_decorated_ curvature profile.

The explanation for the superscripts is the following. The superscript $\bf f$
stands for _face_ , because a singular point of an orbifold with $\mu_{i}=4$
and $n_{i}=3$ may lie in the middle of a face of the orbitriangulation. The
superscript $\bf e$ stands for _edge_ , because a point of the orbifold with
$\mu_{i}=3$ and $n_{i}=2$ may lie in the middle of an edge of the
orbitriangulation. Of course, both points may also lie at a vertex.

On the other hand, a singular point without a superscript must lie at a
vertex. This is because first, the only possible non-vertex points in a
triangulation, with non-trivial stabilizers under an automorphism group, are
the center of an edge and the center of a face, with stabilizers of size $2$
and $3$ respectively. So $n$ must be $2$ or $3$. Second, if $n(6-\mu)\neq 6$,
the point upstairs has positive curvature and hence must be a vertex.

Thus, the justification for this alternative notation is that the
orbitriangulations in a given stratum of flat cone orbifolds depend only on
the decorated curvature profile—that is, which $\mu_{i}$ are decorated with an
${\bf f}$ or ${\bf e}$, see Proposition 2.17 for a more precise statement.

######  Definition 2.16.

The undecorated profile $\mu^{u}$ of a quotient profile $\mu$ is the result of
deleting the decorations ${\bf f}$, ${\bf e}$.

Undecorating corresponds to forgetting the orbifold structure, i.e. taking the
coarse space or resetting $n_{i}=1$. The orbitriangulations with curvature
profile $\mu^{u}$ are simply triangulations in the curvature stratum
$\mu^{u}$.

######  Proposition 2.17.

Let $\kappa$ be an (undecorated) curvature profile. There is a Hermitian
Eisenstein sublattice $\Lambda_{\kappa}\subset\Lambda$ of signature $(1,{\rm
len}(\kappa)-3)$ and a subgroup $\Gamma_{\kappa}\subset{\rm
Stab}_{\Gamma}(\Lambda_{\kappa})$ for which
$\Gamma_{\kappa}\backslash\Lambda_{\kappa}^{+}$ is in bijection with convex
triangulations of the sphere, whose curvature profile is a collision of
$\kappa$.

Now let $\mu$ be a decorated curvature profile and let $\mu^{u}$ be is
undecoration. There is an Eisenstein module

$\Lambda_{\mu^{u}}\subset\Lambda_{\mu}\subset\tfrac{1}{2(1+\zeta_{6})}\Lambda_{\mu^{u}}$

of signature $(1,{\rm len}(\mu)-3)$ and a subgroup $\Gamma_{\mu}\subset{\rm
Stab}_{\Gamma}(\Lambda_{\mu})$ for which
$\Gamma_{\mu}\backslash\Lambda_{\mu}^{+}$ is in bijection with
orbitriangulations with profile a collision of $\mu$.

###### Proof.

First, we treat the undecorated case. In the local period coordinates
$(\int_{p_{j}}^{p_{j+1}}\omega)\in\mathbb{C}^{1,9}$, the condition that
$\kappa_{i}$ singularities $p_{1},\dots,p_{\kappa_{i}}$ have coalesced is
locally described by the equation

$\textstyle\int_{p_{j}}^{p_{j+1}}\omega=0\textrm{ for
}j=1,\dots,\kappa_{i}-1.$

This is a $\mathbb{Z}[\zeta_{6}]$-linear condition, and thus cuts out a
sublattice $\Lambda_{\kappa_{i}}\subset\Lambda$ exactly corresponding to
triangulations with a point of curvature $\kappa_{i}$. These linear conditions
can be imposed for each coalescence, and are independent, giving a sublattice

$\Lambda_{\kappa}:=\cap_{i}\,\Lambda_{\kappa_{i}}\subset\Lambda.$

In fact, it is shown in Thurston’s paper that the completed moduli space
$\overline{\mathcal{M}}_{\kappa_{1}\pi/3\,,\,\dots\,,\,\kappa_{m}\pi/3}$ is a
subball quotient $\mathbb{P}\Gamma_{\kappa}\backslash\mathbb{CH}^{{\rm
len}(\kappa)-3}\subset\mathbb{P}\Gamma\backslash\mathbb{CH}^{9}$. The
bijection between $\Gamma_{\kappa}\backslash\Lambda^{+}_{\kappa}$ and
triangulations works in the same way as for $M$: the triangulation associated
to $v\in\Lambda_{\kappa}^{+}$ corresponds to

$\mathbb{C}v\in\mathbb{P}\\{x\in\Lambda_{\kappa}\otimes_{\mathbb{Z}[\zeta_{6}]}\mathbb{C},\big{|}\,x\star
x>0\\}=\mathbb{CH}^{{\rm len}(\kappa)-3}.$

We now consider the decorated case $\mu$. The moduli space of flat cone
orbifolds containing orbitriangulations with curvature profile a collision of
$\mu$ is simply
$\overline{\mathcal{M}}_{\mu_{1}\pi/3\,,\,\dots\,,\,\mu_{r}\pi/3}$ as in the
undecorated case. Triangulations with curvature profile $\mu^{u}$ correspond
to $\Gamma_{\mu^{u}}$ orbits of $v\in\Lambda_{\mu^{u}}^{+}$ by the previous
paragraph. In period coordinates, admitting a triangulation is simply the
condition $\int_{p_{j}}^{p_{j+1}}\omega\in\mathbb{Z}[\zeta_{6}]\textrm{ for
all }j$.

Let $\mu_{i}$ be an undecorated curvature. Admitting an orbitriangulation
amounts to the slightly weaker period conditions:

(2) $\displaystyle\begin{aligned}
\textstyle\int_{p_{i}}^{p_{j}}\omega&\in\tfrac{1}{1+\zeta_{6}}\mathbb{Z}[\zeta_{6}]\textrm{
if }\mu_{j}=4^{\bf f},\\\
\textstyle\int_{p_{i}}^{p_{j}}\omega&\in\tfrac{1}{2}\mathbb{Z}[\zeta_{6}]\textrm{
if }\mu_{j}=3^{\bf e},\\\
\textstyle\int_{p_{i}}^{p_{j}}\omega&\in\mathbb{Z}[\zeta_{6}]\textrm{
otherwise.}\end{aligned}$

This is because $\frac{1}{1+\zeta_{6}}\mathbb{Z}[\zeta_{6}]$ is the
superlattice of $\mathbb{Z}[\zeta_{6}]$ containing the centers of the standard
triangular tiles of $\mathbb{C}$ (and the vertices), while
$\frac{1}{2}\mathbb{Z}[\zeta_{6}]$ is the superlattice of
$\mathbb{Z}[\zeta_{6}]$ containing the centers of the edges (and the
vertices).

Conditions (2) on period integrals describe a finite index enlargement
$\Lambda_{\mu}\supset\Lambda_{\mu^{u}}$ which is contained in
$\frac{1}{2(1+\zeta_{6})}\Lambda_{\mu^{u}}$. Equality of orbitriangulations is
the same as equality of the underlying flat cone spheres, preserving the
ramification orders $n_{i}$. Thus, $\Gamma_{\mu}\subset\Gamma_{\mu^{u}}$ is a
finite index subgroup. ∎

######  Remark 2.18.

We will identify properties of the Eisenstein lattices/modules
$\Lambda_{\kappa}$ and $\Lambda_{\mu}$ in greater detail in the following
section.

We call $\Lambda_{\mu}$ an Eisenstein module, rather than an Eisenstein
lattice, for the following reason: Its natural Hermitian inner product $\star$
need not be valued in $\mathbb{Z}[\zeta_{6}]$. Unlike $\Lambda_{\mu^{u}}$,
there may be vectors $v\in\Lambda_{\mu}$ for which $v\star v\notin
3\mathbb{Z}$, because an orbitriangulation need not have an even integer
number of triangles.

######  Definition 2.19.

Let $f_{\mu}(q)=c_{0}+\sum_{n>0}c_{n}q^{n}$, where
$n\in\tfrac{1}{6}\mathbb{Z}$, be the completed generating function for
orbitriangulations in the decorated stratum $\mu$: Concretely, for $n>0$,
$c_{n}$ is the count of orbitriangulations
$v\in\Gamma_{\mu}\backslash\Lambda_{\mu}^{+}$ with $2n$ triangles, weighted by
$|{\rm Stab}_{\Gamma_{\mu}}(v)|^{-1}$.

We define the constant term $c_{0}$ in the next section.

######  Example 2.20.

We define the important generating function

$A(q):=\tfrac{1}{6}\sum_{v\in\mathbb{Z}[\zeta_{6}]}q^{v\overline{v}}.$

For any $0$-dimensional decorated profile (i.e. ${\rm len}(\mu)=3$), we have
that $f_{\mu}(q)=A(q^{r})$ where $r\in\tfrac{1}{6}\mathbb{Z}$ is the (possibly
fractional) number of triangles in the smallest area orbitriangulation
$\mathcal{T}_{0}$. This is because every orbitriangulation with decorated
profile $\mu$ is the result of scaling $\mathcal{T}_{0}$ the flat structure by
some nonzero Eisenstein integer.

######  Definition 2.21.

Let ${\rm BB}_{(G,\mu)}(q)$ to be generating function for naive enumeration of
triangulations in the bistratum $(G,\mu)$ i.e. triangulations with
automorphism group $G$ and decorated quotient profile $\mu$.

Note that here we are counting the upstairs triangulations, not the quotient
orbitriangulations, so ${\rm BB}_{(G,\mu)}(q)\in\mathbb{Z}[[q]]$.

######  Proposition 2.22.

Let $\\{M_{(H,\nu)}\\}$ be the collection of all lower-dimensional bistrata in
the closure $\overline{M}_{(G,\mu)}$ of a fixed bistratum. There exist
constants $c_{(H,\nu)}\in\mathbb{Q}$ and $c_{0}\in\mathbb{R}$ for which

$f_{\mu}(q^{|G|})={\rm BB}_{(G,\mu)}(q)+c_{0}+\sum_{(H,\nu)}c_{(H,\nu)}{\rm
BB}_{(H,\nu)}(q).$

###### Proof.

By Corollary 2.8, a triangulation $\mathcal{T}$ with automorphism group $G$
gives rise to a flat cone sphere $\mathcal{T}/G$ admiting an orbitriangulation
with decorated profile $\mu$. But conversely, given a group $G$ and an
orbitriangulation with decorated profile $\mu$, the $G$-cover
$\mathbb{P}^{1}\to\mathbb{P}^{1}/G$ branched over the appropriate orbipoints
gives a triangulation with a $G$-action.

Thus, triangulations in the open bistratum $M_{(G,\mu)}$ with $|G|n$ triangles
are in bijection with orbitriangulations with decorated profile $\mu$ and $n$
triangles, whose $G$-cover has automorphism group $G$. Furthermore, in both
$f_{\mu}(q)$ and ${\rm BB}_{(G,\mu)}(q)$, these (orbi)triangulations are
counted with weight $1$.

In fact, whenever the automorphism group of an orbitriangulation jumps, so
does the automorphism group of its $G$-cover, and the same holds for
singularity collisions. Thus, the $G$-covers of orbitriangulations which
contribute a fixed mass to $f_{\mu}(q)$ lie in a disjoint union of bistrata
$M_{(H,\nu)}$. The proposition follows. ∎

######  Remark 2.23.

The converse to the last paragraph of Proposition 2.22 is false. For instance,
it is possible that an orbitriangulation contributes $1$ to $f_{\mu}(q)$ but
its $G$-cover has an automorphism group larger than $G$, and hence would
contribute less than $1$ in a weighted count. This can occur if one quotients
a triangulation $\mathcal{T}$ by a non-normal subgroup of ${\rm
Aut}^{+}(\mathcal{T})$.

######  Corollary 2.24.

${\rm BB}_{(G,\mu)}(q)$ is a linear combination of $f_{\nu}(q^{|H|})$ ranging
over all bistrata $M_{(H,\nu)}\subset\overline{M}_{(G,\mu)}$ (plus a
constant).

###### Proof.

This follows from Proposition 2.22, which shows that ${\rm BB}_{(G,\mu)}(q)$
and $f_{\mu}(q^{|G|})$ differ by a strictly triangular change-of-basis. ∎

######  Theorem 2.25.

The naive generating function ${\rm BB}(q)$ for fullerenes is a linear
combination of $f_{\mu}(q^{|G|})$, ranging over all bistrata
$M_{(G,\mu)}\subset M$.

###### Proof.

This follows immediately from Corollary 2.24 as

${\rm BB}(q)=\sum_{\begin{subarray}{c}(G,\mu)\textrm{ s.t.}\\\
\kappa\,=\,1^{12}\end{subarray}}{\rm BB}_{(G,\mu)}(q)$

where $\kappa$ is the curvature profile of the $G$-cover of an
orbitriangulation with decorated curvature profile $\mu$. ∎

The final computation for this section is the enumeration of all bistrata.
This is accomplished relatively easily by considering the generic flat cone
quotient for each group:

  1. $I$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{5,3,2}$ with $\mu=\\{5,4^{\bf f},3^{\bf
e}\\}$ the only bistratum.

  2. $O$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{4,3,2}$ with $\mu=\\{5,4^{\bf f},3^{\bf
e}\\}$ the only triangulated bistratum.

  3. $T$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{3,3,2}$ with $\mu=\\{4^{\bf f},4^{\bf
f},3^{\bf e},1\\}$ the largest bistratum.

  4. $D_{6}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{6,2,2}$ with $\mu=\\{5,3^{\bf e},3^{\bf
e},1\\}$ the largest bistratum.

  5. $D_{5}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{5,2,2}$ with $\mu=\\{5,3^{\bf e},3^{\bf
e},1\\}$ the largest bistratum.

  6. $D_{4}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{4,2,2}$ with $\mu=\\{5,3^{\bf e},3^{\bf
e},1\\}$ the largest bistratum.

  7. $D_{3}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{3,2,2}$ with $\mu=\\{4^{\bf f},3^{\bf
e},3^{\bf e},1,1\\}$ the largest bistratum.

  8. $C_{3}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{3,3}$ with $\mu=\\{4^{\bf f},4^{\bf
f},1^{4}\\}$ the largest bistratum.

  9. $D_{2}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{2,2,2}$ with $\mu=\\{3^{\bf e},3^{\bf
e},3^{\bf e},1^{3}\\}$ the largest bistratum.

  10. $C_{2}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}_{2,2}$ with $\mu=\\{3^{\bf e},3^{\bf
e},1^{6}\\}$ the largest bistratum.

  11. $C_{1}$:

$\mathbb{P}^{1}/G=\mathbb{P}^{1}$ with $\mu=\\{1^{12}\\}$ the largest
bistratum.

######  Remark 2.26.

The cuboctahedron is another flat cone sphere in $M$ with $O$ symmetry, but it
cannot be equilaterally triangulated.

All other bistrata result from colliding some singularities of curvature
$\mu_{i}=1$ either into each other or the orbipoints of $\mathbb{P}^{1}/G$.
The results of such collisions are shown in Table LABEL:bistrata. On occasion,
the $G$-cover of an orbitrianguation with specified $(G,\mu)$ automatically
has a larger automorphism group, leading to some repeated bistrata in Table
LABEL:bistrata. This is indicated in the “Description” column, and only occurs
for bistrata of dimension $0$ or $1$.

######  Proposition 2.27.

There are 101 bistrata of $M$.

###### Proof.

The proposition follows from direct enumeration. ∎

Table 1. Bistrata of $M$ $G$ | $|G|$ | $n_{i}$ | $\mu_{i}$ | Weight | Level | $\mathbb{C}f_{\mu}(q^{|G|})$ | Description
---|---|---|---|---|---|---|---
$I$ | $60$ | ${5,3,2}$ | ${5,4^{\bf f},3^{\bf e}}$ | 1 | $30$ | $A(q^{10})$ | icosahedron
$O$ | $24$ | ${4,3,2}$ | ${5,4^{\bf f},3^{\bf e}}$ | 1 | $12$ | $A(q^{4})$ | octahedron
$T$ | $12$ | ${3,3,2}$ | ${4^{\bf f},4^{\bf f},3^{\bf e},1}$ | 2 | $(6,12)$ |  |
|  |  | ${4^{\bf f},4^{\bf f},4}$ | 1 | $12$ | $A(q^{4})$ | $(O,\\{5,4^{\bf f},3^{\bf e}\\})$
|  |  | ${5,4^{\bf f},3^{\bf e}}$ | 1 | $6$ | $A(q^{2})$ | tetrahedron
$D_{6}$ | $12$ | ${6,2,2}$ | ${5,3^{\bf e},3^{\bf e},1}$ |  $2$ (Q) | $6$ | $E_{2}(q^{6})$ |  $6$-tubes
|  |  | ${5,4,3^{\bf e}}$ | $1$ | $18$ | $A(q^{6})$ | doubled hexagon
$D_{5}$ | $10$ | ${5,2,2}$ | ${5,3^{\bf e},3^{\bf e},1}$ |  $2$ (Q) | $5$ | $E_{2}(q^{5})$ |  $5$-tubes
|  |  | ${5,4,3^{\bf e}}$ | $1$ | $15$ | $A(q^{5})$ | pentagonal bipyramid
$D_{4}$ | $8$ | ${4,2,2}$ | ${5,3^{\bf e},3^{\bf e},1}$ |  $2$ (Q) | $4$ | $E_{2}(q^{4})$ |  $4$-tubes
|  |  | ${5,4,3^{\bf e}}$ | $1$ | $12$ | $A(q^{4})$ | $(O,\\{5,4^{\bf f},3^{\bf e}\\})$
$D_{3}$ | $6$ | ${3,2,2}$ | ${4^{\bf f},3^{\bf e},3^{\bf e},1,1}$ | $3$ | $(6,6)$ |  |
|  |  | ${4^{\bf f},3^{\bf e},3^{\bf e},2}$ |  $2$ (Q) | $(6,6)$ |  |
|  |  | ${4^{\bf f},4,3^{\bf e},1}$ | $2$ | $(6,6)$ |  |
|  |  | ${4^{\bf f},4,4}$ | $1$ | $18$ | $A(q^{6})$ | $(D_{6},\\{5,4,3^{\bf e}\\})$
|  |  | ${5,3^{\bf e},3^{\bf e},1}$ |  $2$ (Q) | $3$ | $E_{2}(q^{3})$ |  $3$-tubes
|  |  | ${5,4,3^{\bf e}}$ | 1 | $9$ | $A(q^{3})$ | triangular bipyramid
|  |  | ${4^{\bf f},5,3^{\bf e}}$ | 1 | $3$ | $A(q)$ | doubled triangle
$C_{3}$ | $3$ | ${3,3}$ | ${4^{\bf f},4^{\bf f},1^{4}}$ | $4$ | $(3,3)$ |  |
|  |  | ${4^{\bf f},4^{\bf f},2,1,1}$ | $3$ | $(3,3)$ |  |
|  |  | ${4^{\bf f},4^{\bf f},2,2}$ |  $2$ (Q) | $(3,3)$ |  |
|  |  | ${4^{\bf f},4^{\bf f},3,1}$ | $2$ | $(6,3)$ |  |
|  |  | ${4^{\bf f},4^{\bf f},4}$ | $1$ | $3$ | $A(q)$ | $(D_{3},\\{4^{\bf f},5,3^{\bf e}\\})$
|  |  | ${5,4^{\bf f},1^{3}}$ | $3$ | $(3,3)$ |  |
|  |  | ${5,4^{\bf f},2,1}$ |  $2$ (Q) | $(3,3)$ |  |
|  |  | ${5,4^{\bf f},3}$ | $1$ | $6$ | $A(q^{2})$ | $(T,\\{5,4^{\bf f},3^{\bf e}\\})$
|  |  | ${5,5,1,1}$ |  $2$ (Q) | $3$ | $E_{2}(q^{3})$ | $(D_{3},\\{5,3^{\bf e},3^{\bf e},1\\})$
|  |  | ${5,5,2}$ | $1$ | $9$ | $A(q^{3})$ | $(D_{3},\\{5,4,3^{\bf e}\\})$
$D_{2}$ | $4$ | ${2,2,2}$ | ${3^{\bf e},3^{\bf e},3^{\bf e},1^{3}}$ | $4$ | $(2,4)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},3^{\bf e},2,1}$ | $3$ | $(6,4)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},3^{\bf e},3}$ |  $2$ (Q) | $2$ | $E_{2}(q^{2})$ | pillowcases
|  |  | ${4,3^{\bf e},3^{\bf e},1,1}$ | $3$ | $(6,4)$ |  |
|  |  | ${4,3^{\bf e},3^{\bf e},2}$ |  $2$ (Q) | $(6,4)$ |  |
|  |  | ${4,4,3^{\bf e},1}$ | $2$ | $(6,4)$ |  |
|  |  | ${4,4,4}$ | $1$ | $12$ | $A(q^{4})$ | $(O,\\{5,4^{\bf f},3^{\bf e}\\})$
|  |  | ${5,3^{\bf e},3^{\bf e},1}$ |  $2$ (Q) | $2$ | $E_{2}(q^{2})$ |  $2$-tubes
|  |  | ${5,4,3^{\bf e}}$ | $1$ | $6$ | $A(q^{2})$ | doubled rhombus
$C_{2}$ | $2$ | ${2,2}$ | ${3^{\bf e},3^{\bf e},1^{6}}$ | $6$ | $(2,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},2,1^{4}}$ | $5$ | $(6,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},2,2,1,1}$ | $4$ | $(6,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},2,2,2}$ | $3$ | $(6,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},3,1^{3}}$ | $4$ | $(2,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},3,2,1}$ | $3$ | $(6,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},3,3}$ |  $2$ (Q) | $2$ | $E_{2}(q^{2})$ | $(D_{2},\\{3^{\bf e},3^{\bf e},3^{\bf e},3\\})$
|  |  | ${3^{\bf e},3^{\bf e},4,1,1}$ | $3$ | $(6,2)$ |  |
|  |  | ${3^{\bf e},3^{\bf e},4,2}$ |  $2$ (Q) | $3$ |  |
|  |  | ${3^{\bf e},3^{\bf e},5,1}$ |  $2$ (Q) | $1$ | $E_{2}(q)$ |  $1$-tubes
|  |  | ${4,3^{\bf e},1^{5}}$ | $5$ | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},2,1^{3}}$ | $4$ | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},2,2,1}$ | $3$ | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},3,1,1}$ | $3$ | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},3,2}$ |  $2$ (Q) | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},4,1}$ | $2$ | $(6,2)$ |  |
|  |  | ${4,3^{\bf e},5}$ | $1$ | $3$ | $A(q)$ | doubled pediment
|  |  | ${4,4,1^{4}}$ | $4$ | $(3,2)$ |  |
|  |  | ${4,4,2,1,1}$ | $3$ | $(3,2)$ |  |
|  |  | ${4,4,2,2}$ |  $2$ (Q) | $(3,2)$ |  |
|  |  | ${4,4,3,1}$ | $2$ | $(6,2)$ |  |
|  |  | ${4,4,4}$ | $1$ | $6$ | $A(q^{2})$ | $(D_{2},\\{5,4,3^{\bf e}\\})$
|  |  | ${5,3^{\bf e},1^{4}}$ | $4$ | $(2,2)$ |  |
|  |  | ${5,3^{\bf e},2,1,1}$ | $3$ | $(6,2)$ |  |
|  |  | ${5,3^{\bf e},2,2}$ | $2$ | $(6,2)$ |  |
|  |  | ${5,3^{\bf e},3,1}$ |  $2$ (Q) | $(2,2)$ |  |
|  |  | ${5,3^{\bf e},4}$ | $1$ | $3$ | $A(q)$ | $(D_{3},\\{4^{\bf f},5,3^{\bf e})$
|  |  | ${5,4,1^{3}}$ | $3$ | $(3,2)$ |  |
|  |  | ${5,4,2,1}$ |  $2$ (Q) | $(3,2)$ |  |
|  |  | ${5,4,3}$ | $1$ | $12$ | $A(q^{4})$ | doubled dart
|  |  | ${5,5,1,1}$ |  $2$ (Q) | 2 | $E_{2}(q^{2})$ | $(D_{2},\\{5,3^{\bf e},3^{\bf e},1\\})$
|  |  | ${5,5,2}$ | $1$ | $6$ | $A(q^{2})$ | $(D_{2},\\{5,4,3^{\bf e}\\})$
$C_{1}$ | $1$ | $\mathbb{P}^{1}$ | $1^{12}$ | $10$ | $1$ | $E_{10}(q)$ | buckyballs
|  |  | $2,1^{10}$ | $9$ | $3$ |  |
|  |  | ${3,1^{9}}$ | $8$ | $2$ |  |
|  |  | ${2,2,1^{9}}$ | $8$ | $3$ |  |
|  |  | ${3,2,1^{7}}$ | $7$ | $6$ |  |
|  |  | $\cdots$ | $\cdots$ | $6$ |  |
|  |  | $3,3,3,3$ |  $2$ (Q) | $2$ | $E_{2}(q^{2})$ | $(D_{2},\\{3^{\bf e},3^{\bf e},3^{\bf e},3\\})$
|  |  | $4,3,3,2$ |  $2$ (Q) | $6$ |  |
|  |  | $4,4,2,2$ |  $2$ (Q) | $3$ |  | $(C_{2},\\{3^{\bf e},3^{\bf e},4,2\\})$
|  |  | $4,4,3,1$ | $2$ | $6$ |  |
|  |  | $5,3,2,2$ | $2$ | $6$ |  |
|  |  | $5,3,3,1$ |  $2$ (Q) | $2$ |  |
|  |  | $5,4,2,1$ |  $2$ (Q) | $3$ |  |
|  |  | $5,5,1,1$ |  $2$ (Q) | $1$ | $E_{2}(q)$ | $(C_{2},\\{3^{\bf e},3^{\bf e},5,1\\})$
|  |  | $4,4,4$ | $1$ | $3$ | $A(q)$ | $(D_{3},\\{4^{\bf f},5,3^{\bf e}\\})$
|  |  | $5,4,3$ | $1$ | $6$ | $A(q^{2})$ | (doubled 30-60-90)
|  |  | $5,5,2$ | $1$ | $3$ | $A(q)$ | $(C_{2},\\{4,3^{\bf e},5\\})$

Taxonomy of triangulations

We explain the terms used in the “Description” column of Table LABEL:bistrata.

######  Definition 2.28.

A doubled polygon is the flat cone sphere constructed by taking two copies of
the polygon (with opposite orientations), and gluing them along their common
boundary to form a sphere, with cone points at the vertices of the glued
polygons.

Doubles appearing as $0$-dimensional bistrata are of the (regular) hexagon,
(equilateral) triangle, the $\mathit{30}$-$\mathit{60}$-$\mathit{90}$
triangle, the $60$-$120$-$60$-$120$ rhombus, the pediment which is a
$30$-$30$-$120$ triangle, and the dart which is a $60$-$90$-$120$-$90$
quadrilateral forming a fundamental domain for $\triangle/C_{3}$.

######  Definition 2.29.

The pillowcases are the quotients by $\\{\pm 1\\}$ of equilaterally
triangulated elliptic curves $\mathbb{C}/L$, $L\subset\mathbb{Z}[\zeta_{6}]$.

######  Definition 2.30.

A $k$-gonal bipyramid is the result of gluing two pyramids with $k$-gonal base
along their bases. The resulting triangulation has $2k$ triangles and $D_{k}$
symmetry (at least).

The hexagonal bipyramid is also a doubled hexagon, and the square bipyramid is
also an octahedron.

######  Definition 2.31.

A $k$-tube, $k\in\\{1,2,3,4,5,6\\}$ is built from three pieces. At the left
and right ends, we have a $k$-gonal pyramid with equilateral triangular faces.
These $k$-gonal pyramids are glued to the two boundary components of a flat
cylinder, which goes between the two ends.

The $k$-tubes form $1$-dimensional bistrata for the group $D_{k}$. The
terminology is motivated by buckytubes, which are fullerenes dual to
$5$-tubes.

##  3\. Modularity of bistratum generating functions

In this section, we prove the modularity properties of the generating
functions $f_{\mu}(q^{|G|})$. We recall from [10], see also Allcock [2], that
the lattice $(\Lambda,\star)$ is $(1+\zeta_{6})$-modular:
$(1+\zeta_{6})\Lambda^{\vee}=\Lambda$ where

$\Lambda^{\vee}:=\\{v\in\Lambda\otimes_{\mathbb{Z}[\zeta_{6}]}\mathbb{C}\,\big{|}\,v\star
w\in\mathbb{Z}[\zeta_{6}]\textrm{ for all }w\in\Lambda\\}$

is the Hermitian dual. It is the unique such Eisenstein lattice of signature
$(1,9)$. We have ${\rm Re}(v\star w)\in\tfrac{3}{2}\mathbb{Z}$, and so we have
an even $\mathbb{Z}$-lattice $(\Lambda,\cdot)$ of signature $(2,18)$ defined
by

$v\cdot w:=\tfrac{2}{3}{\rm Re}(v\star w).$

This lattice is even unimodular of signature $(2,18)$, hence isometric
$I\\!I_{2,18}$. Denote the $\mathbb{Z}$-dual of $(\Lambda,\cdot)$ by
$\Lambda^{*}$. Then $\Lambda^{*}=\Lambda$. For any even $\mathbb{Z}$-lattice,
$\cdot$ defines a $\mathbb{Q}/2\mathbb{Z}$-valued quadratic form on the
discriminant group $\Lambda^{*}/\Lambda$.

######  Definition 3.1.

We define $\Gamma(N)\subset SL_{2}(\mathbb{Z})$ to be the subgroup of $2\times
2$ matrices which reduce to the identity mod $N$. It is contained in the group
$\Gamma_{1}(N)$ of matrices which are strictly upper triangular mod $N$. Given
any subgroup $H\subset(\mathbb{Z}/N\mathbb{Z})^{*}$, we define $\Gamma_{H}(N)$
as the matrices whose reduction mod $N$ is upper triangular, and with diagonal
entries in $H$.

######  Definition 3.2.

A real-analytic function $f\colon\mathbb{H}\to\mathbb{C}$ is modular of weight
$(r,s)$ for a subgroup $\Gamma\subset SL_{2}(\mathbb{Z})$ if it satisfies

$f(\gamma\cdot\tau)=(c\tau+d)^{r}(c\overline{\tau}+d)^{s}f(\tau)$

for all $\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma$.

We will need a generalization of this to vector-valued modular forms:

######  Definition 3.3.

Let $\rho\colon SL_{2}(\mathbb{Z})\to GL(E)$ be a finite-dimensional
representation on a complex vector space $E$. A real-analytic function
$f\colon\mathbb{H}\to E$ is modular for $SL_{2}(\mathbb{Z})$ of weight $(r,s)$
for the representation $\rho$ if

$f(\gamma\cdot\tau)=(c\tau+d)^{r}(c\overline{\tau}+d)^{s}\rho(\gamma)\cdot
f(\tau)$

for all $\gamma\in SL_{2}(\mathbb{Z})$.

######  Definition 3.4.

A modular form of weight $k$ is a function $f(\tau)$ on the upper half-plane
such that:

  1. (1)

$f$ is holomorphic,

  2. (2)

$f$ is modular of weight $(k,0)$, and

  3. (3)

$f$ has bounded growth at the cusps of $\mathbb{H}$.

This definition applies both to modular forms valued in a representation
$\rho$ and scalar-valued modular forms for subgroups of $SL_{2}(\mathbb{Z})$.

Thus, we distinguish between “modular” functions and “modular forms.”

######  Definition 3.5.

Let $(\Lambda,\cdot)$ be an even $\mathbb{Z}$-lattice of even rank and
signature $(r,s)$. The Weil representation

$\rho_{\Lambda}\colon SL_{2}(\mathbb{Z})\to
GL(\mathbb{C}[\Lambda^{*}/\Lambda])$

sends generators $S:=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$ and
$T:=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\in SL_{2}(\mathbb{Z})$ to

$\displaystyle\rho_{\Lambda}(S)e_{\gamma}$
$\displaystyle=\frac{i^{(s-r)/2}}{\sqrt{|\Lambda^{*}/\Lambda|}}\sum_{\delta\in\Lambda^{*}/\Lambda}e^{-2\pi
i(\gamma\cdot\delta)}e_{\delta}$ $\displaystyle\rho_{\Lambda}(T)e_{\gamma}$
$\displaystyle=e^{\pi i(\gamma\cdot\gamma)}e_{\gamma}.$

Here $\gamma\in\Lambda^{*}/\Lambda$ and $e_{\gamma}$ is a basis element of the
group algebra $\mathbb{C}[\Lambda^{*}/\Lambda]$.

######  Definition 3.6.

Let $(\Lambda,\cdot)$ be a $\mathbb{Z}$-lattice of signature $(2r,2s)$, and
let $\mathbb{D}:=Gr^{+}(2r,V\otimes\mathbb{R})$ be the Grassmannian of
positive-definite $2r$-planes. The Siegel theta correspondence is the function
$\mathbb{D}\times\mathbb{H}\to\mathbb{C}[\Lambda^{*}/\Lambda]$ given by

$\Theta(p,\tau):=\sum_{v\in\Lambda^{*}}q^{\frac{1}{2}v^{+}\cdot
v^{+}}\overline{q}^{-\frac{1}{2}v^{-}\cdot v^{-}}e_{[v]}.$

Here $q=e^{2\pi i\tau}$, $v^{+}$ and $v^{-}$ are orthogonal projections to the
positive $2r$-space $p\in\mathbb{D}$ and its negative-definite perpendicular
$2s$-space $p^{\perp}$ respectively, and $e_{[v]}$ is the basis element of
$\mathbb{C}[\Lambda^{*}/\Lambda]$ corresponding to the reduction of
$v\in\Lambda^{*}$.

Let $O^{*}(\Lambda)$ be the subgroup of the orthogonal group acting trivially
on $\Lambda^{*}/\Lambda$. It is clear from the definition that $\Theta(g\cdot
p,\tau)=\Theta(p,\tau)$ for all $g\in O^{*}(\Lambda)$. We additionally have
the transformation property

$\Theta\left(p,\frac{-1}{\tau}\right)=\tau^{r}\overline{\tau}^{s}\rho_{\Lambda}(S)\cdot\Theta(p,\tau)$

in $\tau$, which follows from the Poisson summation formula. Thus, in the
$\tau$ variable, $\Theta(p,\tau)$ is modular of weight $(r,s)$ for the Weil
representation $\rho_{\Lambda}$.

######  Definition 3.7.

Let $(\Lambda,\star)$ be an Eisenstein lattice of signature $(1,s)$ with
$\star$ valued in $(1+\zeta_{6})\mathbb{Z}[\zeta_{6}]$ and let $v\cdot
w:=\tfrac{2}{3}{\rm Re}(v\star w)$ be the corresponding even
$\mathbb{Z}$-lattice of signature $(2,2s)$. Let $U^{*}(\Lambda)$ be the
subgroup of the unitary group acting trivially on $\Lambda^{*}/\Lambda$.
Define the function

$g_{\Lambda}(\tau):=\frac{({\rm
Im}\,\tau)^{s}}{|Z(U^{*}(\Lambda))|}\int_{U^{*}(\Lambda)\backslash\mathbb{CH}^{s}}\Theta(p,\tau)\,dp$

which is a real-analytic function from $\mathbb{H}$ to
$\mathbb{C}[\Lambda^{*}/\Lambda]$, modular of weight $(1-s,0)$ with respect to
the Weil representation.

######  Theorem 3.8.

Let $\Gamma\subset U^{*}(\Lambda)$ be a finite index subgroup. If $s>1$ or
$s=1$ and $\Gamma\backslash\mathbb{CH}^{s}$ is compact, then

$f_{\Lambda}(\tau):=[U^{*}(\Lambda):\Gamma]\left(\frac{1}{2\pi
i}\partial_{\tau}\right)^{s}g_{\Lambda}(\tau)=c_{0}+\sum_{\begin{subarray}{c}v\in\Gamma\backslash\Lambda^{*}\\\
v\cdot v>0\end{subarray}}\frac{1}{|{\rm
Stab}_{\Gamma}(v)|}q^{\frac{1}{2}v\cdot v}e_{[v]}$

and furthermore, this is a modular form of weight $1+s$ for
$SL_{2}(\mathbb{Z})$, valued in the Weil representation $\rho_{\Lambda}$.

If $s=1$ and $\Gamma\backslash\mathbb{CH}^{s}$ is non-compact,
$f_{\Lambda}(\tau)$ is a quasimodular form of weight $2$ for
$SL_{2}(\mathbb{Z})$, valued in $\rho_{\Lambda}$—the transformation law for
$\tau\mapsto-1/\tau$ is corrected by adding a factor of the form $c/({\rm
Im}\,\tau)$ for some $c\in\mathbb{C}[\Lambda^{*}/\Lambda]$.

###### Sketch..

This is essentially due to Siegel [24], and is proved in greater generality by
Kudla and Millson in [14]; the details are worked out in the unimodular case
for $s>1$ in [10]. The vector-valued case follows the same proof.

The only difficulty arises in the $s=1$ case, see [10, bottom of p. 18].
Essentially, if $\Gamma\backslash\mathbb{CH}^{s}$ is $1$-dimensional and has a
cusp, there is only conditional convergence of the integral defining
$g_{\Lambda}(\tau)$. Then, the summation over $v\in\Lambda$ and integration
over $\Gamma\backslash\mathbb{CH}^{s}$ fail to commute, and a more careful
analysis leads to the extra factor of $c/({\rm Im}\,\tau)$. ∎

######  Proposition 3.9.

Let $f\colon\mathbb{H}\to\mathbb{C}[\Lambda^{*}/\Lambda]$ be a modular (resp.
quasimodular) form valued in the Weil representation $\rho_{\Lambda}$ of an
even $\mathbb{Z}$-lattice $\Lambda$ of even dimension. Let $N$ be an integer
for which $\Lambda^{*}/\Lambda$ is $N$-divisible. Then, the
$e_{\gamma}$-coefficient of $f(\tau)$ is a modular (resp. quasimodular) form
for $\Gamma(N)$, for any $\gamma\in\Lambda^{*}/\Lambda$.

###### Proof.

The level of $(\Lambda,\cdot)$ is the smallest positive integer $L$ for which
$L\Lambda^{*}$ is an even lattice. It is well-known that the Weil
representation $\rho_{\Lambda}\colon SL_{2}(\mathbb{Z})\to
GL(\mathbb{C}[\Lambda^{*}/\Lambda])$ factors through
$SL_{2}(\mathbb{Z}/L\mathbb{Z})$. The proposition follows. ∎

######  Definition 3.10.

Let $\kappa=\\{\kappa_{1},\dots,\kappa_{n}\\}$ be a curvature profile. Define
a finite group

$\Delta_{\kappa_{i}}:=\left\\{\begin{array}[]{lll}0&\mbox{if
}\kappa_{i}=1,5\\\ \mathbb{Z}/3\mathbb{Z}&\mbox{if }\kappa_{i}=2,4\\\
\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}&\mbox{if
}\kappa_{i}=3\end{array}\right.$

and $\Delta_{\kappa}:=\bigoplus_{i}\Delta_{\kappa_{i}}$.

An Eisenstein structure on a $\mathbb{Z}$-lattice is an order $3$ isometry
whose only fixed point is $0$. There are Eisenstein structures on $\\{0\\}$,
$A_{2}$, $D_{4}$, $E_{6}$, $E_{8}$ and we call the resulting Eisenstein
lattices

$A_{0}^{\mathbb{Z}[\zeta_{6}]},\,A_{1}^{\mathbb{Z}[\zeta_{6}]},\,A_{2}^{\mathbb{Z}[\zeta_{6}]},\,A_{3}^{\mathbb{Z}[\zeta_{6}]},\,A_{4}^{\mathbb{Z}[\zeta_{6}]}$

because their Dynkin diagrams over $\mathbb{Z}[\zeta_{6}]$ have $A_{n}$
diagram shape: There are $n$ generators $\\{\alpha_{i}\\}$ over
$\mathbb{Z}[\zeta_{6}]$, satisfying

$\displaystyle\alpha_{j}\star\alpha_{j}$ $\displaystyle=-3$
$\displaystyle\alpha_{j}\star\alpha_{j+1}$ $\displaystyle=1+\zeta_{6}\textrm{
for }j=1,\dots,n-1$ $\displaystyle\alpha_{j}\star\alpha_{k}$
$\displaystyle=0\textrm{ for }|j-k|\geq 2.$

The discriminant groups of their underlying $\mathbb{Z}$-lattices $\\{0\\}$,
$A_{2}$, $D_{4}$, $E_{6}$, $E_{8}$ are exactly the groups
$\Delta_{\kappa_{i}}$ for $\kappa_{i}=1,2,3,4,5$ respectively. As the
discriminant of an even lattice, $\Delta_{\kappa_{i}}$ has a
$\mathbb{Q}/2\mathbb{Z}$-valued quadratic form, descended from the extension
of $\cdot$ to $\Lambda^{*}$.

######  Proposition 3.11.

Let $\Lambda_{\kappa}\subset\Lambda$ be the sublattice corresponding to convex
triangulations with curvature profile $\kappa$ (cf. Proposition 2.17). The
disciminant group $\Lambda_{\kappa}^{*}/\Lambda_{\kappa}$ of the underlying
$\mathbb{Z}$-lattice $(\Lambda_{\kappa},\cdot)$ is a subquotient $I^{\perp}/I$
associated to an isotropic (possibly zero) subspace $I\subset\Delta_{\kappa}$
for the $\mathbb{Q}/2\mathbb{Z}$-valued quadratic form on $\Delta_{\kappa}$.

###### Proof.

In the proof of Proposition 2.17, we construct $\Lambda_{\kappa}$ as the
intersection $\cap_{i}\Lambda_{\kappa_{i}}$ where $\Lambda_{\kappa_{i}}$ is
the sublattice for which $\kappa_{i}-1$ periods $\int_{p_{j}}^{p_{j+1}}\omega$
equal zero.

In fact, we claim
$\Lambda_{\kappa_{i}}=(A^{\mathbb{Z}[\zeta_{6}]}_{\kappa_{i}-1})^{\perp}$ for
an embedded copy of $A^{\mathbb{Z}[\zeta_{6}]}_{\kappa_{i}-1}\subset\Lambda$.
The braid half-twisting $p_{j},p_{j+1}$ maps to a triflection in $U(\Lambda)$
along an Eisenstein root $\alpha_{j}$ satisfying
$\alpha_{j}\star\alpha_{j}=-3$. Thus, the locus where $p_{j},p_{j+1}$ have
collided is the perpendicular of $\alpha_{j}$. Since the half-twists of
$p_{j},p_{j+1}$ and $p_{k},p_{k+1}$ commute unless $|j-k|\leq 1$, the
corresponding triflections in $U(\Lambda)$ commute, and so
$\alpha_{j}\star\alpha_{k}=0$ for $|j-k|\geq 2$. Analyzing a braid cyclically
rotating $p_{j},p_{j+1},p_{j+2}$ gives the formula
$\alpha_{j}\star\alpha_{j+1}=1+\zeta_{6}$.

We conclude that

$\Lambda_{\kappa}=\left(\textstyle\bigoplus_{i}A^{\mathbb{Z}[\zeta_{6}]}_{\kappa_{i}-1}\right)^{\perp}$

for some embedding
$\phi\colon\bigoplus_{i}A^{\mathbb{Z}[\zeta_{6}]}_{\kappa_{i}-1}\hookrightarrow\Lambda$.
The Eisenstein root lattices for each coalescence are perpendicular to each
other, because they involve braiding disjoint collections of singularities. In
general, $\phi$ may not be a primitive embedding, and
$\Lambda_{\kappa}^{\perp}$ is its saturation. Finally, observe that
$\Lambda_{\kappa}^{\perp}$ and $\Lambda_{\kappa}$ are mutually perpendicular,
saturated sublattices of the unimodular $\mathbb{Z}$-lattice
$(\Lambda,\cdot)$. So their discriminant groups are canonically isomorphic.
Thus, the discriminant of $(\Lambda_{\kappa},\cdot)$ is the subquotient
associated to an isotropic subspace $I\subset\Delta_{\kappa}$ of the
discriminant of
$\bigoplus_{i}(A^{\mathbb{Z}[\zeta_{6}]}_{\kappa_{i}-1},\cdot)$. ∎

######  Corollary 3.12.

The discriminant group of $(\Lambda_{\kappa},\cdot)$ is:

  1. (1)

$1$-divisible if $\kappa_{i}\in\\{1,5\\}$ for all $i$,

  2. (2)

$2$-divisible if $\kappa_{i}\in\\{1,3,5\\}$ for all $i$,

  3. (3)

$3$-divisible if $\kappa_{i}\in\\{1,2,4,5\\}$ for all $i$,

  4. (4)

$6$-divisible if $\kappa_{i}\in\\{1,2,3,4,5\\}$ for all $i$.

Define $L(\kappa)\in\\{1,2,3,6\\}$ to be the above integer, for which
$\Lambda_{\kappa}$ is automatically $L(\kappa)$-divisible.

######  Theorem 3.13.

The weighted generating function $f_{\kappa}(q)$ for triangulations in the
closure of the curvature stratum associated to $\kappa$ is a (quasi)modular
form for $\Gamma_{1}(L(\kappa))$ of weight ${\rm len}(\kappa-2)$.

###### Proof.

This follows directly from Corollary 3.12, Proposition 3.9, and Theorem 3.8
applied to the $e_{0}$-coefficient, as

$f_{\kappa}(q)=\frac{1}{[\Gamma_{\kappa}:\Gamma]}\cdot[e_{0}]\,f_{\Lambda_{\kappa}}(q).$

Here $\Gamma=U^{*}(\Lambda_{\kappa})\cap\Gamma_{\kappa}$. Note that we get the
congruence subgroup $\Gamma_{1}(L(\kappa))$ rather than $\Gamma(L(\kappa))$
because $f_{\kappa}(\tau)$ is automatically a power series in $q$, rather than
a fractional power of $q$. ∎

######  Remark 3.14.

This sets the constant coefficient $f_{\kappa}(q)$, which was left ambiguous
in Definition 2.19, to be

$c_{0}=\frac{d!{\rm
Vol}(\mathbb{P}\Gamma_{\kappa}\backslash\mathbb{CH}^{d})}{6(4\pi)^{d}}$

where $d={\rm len}(\kappa)-3$ is the dimension.

Finally, we extend these results to the enumeration of orbitriangulations with
decorated profile $\mu$.

######  Proposition 3.15.

The Eisenstein module $\Lambda_{\mu}\supset\Lambda_{\mu^{u}}$ counting
orbitriangulations (cf. Proposition 2.17) corresponds to a partial saturation
of $\Lambda_{\mu^{u}}$ in $\Lambda_{\mu^{u}}^{*}$, saturating each summand of
$\Delta_{\mu}=\bigoplus\Delta_{\mu_{i}}$ with a decoration $3^{\bf e}$ or
$4^{\bf f}$.

###### Proof.

It was observed in the proof of Proposition 2.17 that allowing $p_{j}$ to be
the center of an orbi-face (decoration $4^{\bf f}$) or center of an orbi-edge
(decoration $3^{\bf e}$) corresponds to allowing the period to lie in
$\int_{p_{1}}^{p_{j}}\omega\in\tfrac{1}{1+\zeta_{6}}\mathbb{Z}[\zeta_{6}]$ or
$\tfrac{1}{2}\mathbb{Z}[\zeta_{6}]$, respectively.

Note that
$\tfrac{1}{1+\zeta_{6}}\mathbb{Z}[\zeta_{6}]/\mathbb{Z}[\zeta_{6}]=\mathbb{Z}/3\mathbb{Z}$
and
$\tfrac{1}{2}\mathbb{Z}[\zeta_{6}]/\mathbb{Z}[\zeta_{6}]=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$.
We can exactly identify this weakening of the period integrality with allowing
the period vector to lie in the $\mathbb{Z}/3\mathbb{Z}$\- or
$\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$-enlargement corresponding
to saturating $\Lambda_{\mu^{u}}\subset\Lambda_{\mu_{u}}^{*}$ in the
corresponding summand of $\Delta_{\mu}$. ∎

######  Theorem 3.16.

The weighted generating function $f_{\mu}(q)$ for orbitriangulations with
decorated profile $\mu$ is a modular form for $\Gamma(L(\mu^{u}))$ of weight
${\rm len}(\mu)-2$, possibly quasimodular if ${\rm len}(\mu)=4$.

###### Proof.

The proof is the same as Theorem 3.13, except that we sum over the
coefficients $[e_{\gamma}]\,f_{\Lambda_{\mu}}(q)$ ranging over all
$\gamma\in\Lambda_{\mu}/\Lambda_{\mu^{u}}\subset\Lambda_{\mu^{u}}^{*}/\Lambda_{\mu^{u}}.$
In particular, the resulting series may involve a fractional power of $q$, and
so we only deduce modularity for $\Gamma(L(\mu^{u}))$ as opposed to
$\Gamma_{1}(L(\mu^{u}))$. ∎

######  Remark 3.17.

The $1/|{\rm Aut}^{+}\mathcal{T}|$-weighted counts of generating functions for
orbitriangulations in any fixed stratum of sextic differentials (even for
higher genus curves) is a mixed weight quasimodular form for $\Gamma(6)$ by
[8, 9]. While this bounds the dimension of the space in which ${\rm BB}(q)$
lives, it does not give sufficient control on the dimension for the purposes
of this paper.

######  Proposition 3.18.

Let $h(q)$ be a quasimodular form for $\Gamma(N)$ which is a power series in
$q^{1/d}$ for some divisor $d\mid N$. Then $h(q^{d})\in\mathbb{C}[[q]]$ is a
quasimodular form for $\Gamma_{H}(Nd)$ where
$H\subset(\mathbb{Z}/Nd\mathbb{Z})^{*}$ is the subgroup of units which are $1$
mod $N$.

###### Proof.

Because $h(q)$ is a series in $q^{1/d}$, it is modular for any matrix

$M=\begin{pmatrix}1+a_{1}N&a_{2}d\\\
a_{3}N&1+a_{4}N\end{pmatrix}\in\Gamma_{1}(N).$

Substituting $q\to q^{d}$ corresponds to the substitution $\tau\mapsto d\tau$,
which is induced by the element

$D=\begin{pmatrix}d&0\\\ 0&1\end{pmatrix}\in GL_{2}^{+}(\mathbb{Q}).$

Then $h(q^{d})$ is modular for any matrix

$D^{-1}MD=\begin{pmatrix}1+a_{1}N&a_{2}\\\ a_{3}dN&1+a_{4}N\end{pmatrix}$

which exactly defines the group $\Gamma_{H}(Nd)$. ∎

######  Definition 3.19.

Define $\Gamma(N,d)$ to be $\Gamma_{H}(Nd)$ where $H$ is the subgroup of
$(\mathbb{Z}/Nd\mathbb{Z})^{*}$ of units congruent to $1$ mod $N$. We say that
a (quasi)modular form for $\Gamma(N,d)$ is of level $(N,d)$.

######  Theorem 3.20.

For any bistratum $(G,\mu)$, its generating function for triangulations
$f_{\mu}(q^{|G|})$ is a modular form (quasimodular when ${\rm len}(\mu)=4$)
for $\Gamma(L(\mu),|G|)$ of weight ${\rm len}(\mu)-2$.

This follows directly from Theorem 3.16 and Proposition 3.18.

######  Definition 3.21.

Let $M_{k}^{N,d}=M_{k}(\Gamma(N,d))$ denote the space of modular forms of
level $(N,d)$ and weight $k$ and let $M_{k}^{N}$ denote the space of modular
forms for $\Gamma_{1}(N)=\Gamma(N,1)$.

We have $M_{k}^{N,d}\supset M_{k}^{N^{\prime},d^{\prime}}$ when
$N^{\prime}\mid N$ and $d^{\prime}\mid d$.

######  Proposition 3.22.

The generating function ${\rm BB}(q)$ for oriented fullerenes lies in the
$152$-dimensional vector space

$\displaystyle\mathbb{O}:=M_{10}^{1}\,\oplus$ $\displaystyle\,M_{9}^{3}\oplus
M_{8}^{6}\oplus M_{7}^{6}\oplus M_{6}^{6,2}\oplus M_{5}^{6,2}\oplus
M_{4}^{6,12}\oplus M_{3}^{6,12}\oplus M_{2}^{6,12}\,\oplus$
$\displaystyle\mathbb{C}E_{2}(q)\oplus\mathbb{C}E_{2}(q^{5})\oplus_{n\in\\{1,2,3,4,5,6,10\\}}\mathbb{C}A(q^{n})\oplus\mathbb{C}.$

###### Proof.

For all bistrata of dimension $2$ (weight $3$) or greater, we simply go
through the list in Table LABEL:bistrata, applying Theorem 3.20 in each case,
to determine a space of modular forms containing $f_{\mu}(q^{|G|})$. The
results are listed in the level and weight columns of Table LABEL:bistrata.
Using the containments between $M_{k}^{N,d}$ spaces, we take the least common
multiples of the $N$’s and the $d$’s to produce a level $(N,d)$ in that weight
containing all $f_{\mu}(q^{|G|})$.

In weight $2$, the series $f_{\mu}(q^{|G|})$ may be only a quasimodular form,
indicated in the weight column of Table LABEL:bistrata by (Q). Any
quasimodular form of weight $2$ is modular after subtracting a constant
multiple of $E_{2}(q)=-\tfrac{1}{24}+\sum_{n\geq 1}\sigma(n)q^{n}$. So
$M_{2}^{6,12}+\mathbb{C}E_{2}(q)$ covers all the $1$-dimensional bistrata for
which $f_{\mu}(q^{|G|})$ is not explicitly listed.

The bistrata of dimension $1$ where $f_{\mu}(q^{|G|})$ is given explicitly are
the $k$-tubes. Any $k$-tube is a $C_{k}$-cover of a triangulation in the
curvature stratum $\kappa=\\{5,5,1,1\\}$, branched over the two points of
curvature $5$. We necessarily have $f_{\\{5,5,1,1\\}}(q)\in\mathbb{C}E_{2}(q)$
because it is quasimodular of level $1$ and weight $2$. Thus, the generating
function for $k$-tubes is $E_{2}(q^{k})$, up to a constant. Note
$E_{2}(q)-kE_{2}(q^{k})$ is modular of level $k$ and so lies in $M_{2}^{6,12}$
for $k=1,2,3,4,6$. So the only extra series we need beyond $E_{2}(q)$ is
$E_{2}(q^{5})$.

The bistrata of dimension $0$ correspond to flat cone spheres unique up to
scaling. If the smallest triangulation in this bistratum has $2n$ triangles,
then the generating function for the stratum is $A(q^{n})$ because the
corresponding Eisenstein lattice has rank $1$ (Ex. 2.20). Finally, we add a
constant in $\mathbb{C}$. Then, by Theorem 2.25, the resulting space of forms
contains ${\rm BB}(q)$.

The spaces in the decomposition are mutually linearly independent, since for
weight $\geq 3$, there is only one space of that weight, and for weight $2$,
the only generator beyond $M^{6,12}_{2}\oplus\mathbb{C}E_{2}(q)$ is
$E_{2}(q^{5})$, which does not lie in the former space. The weight $1$ terms
$A(q^{n})$ are also linearly independent. ∎

######  Proposition 3.23.

The map $\mathbb{C}[[q]]\to\mathbb{C}[[q]]/q^{200}\simeq\mathbb{C}^{200}$
which truncates at $q^{200}$ is injective upon restriction to $\mathbb{O}$.

###### Proof.

It is an easy check in SAGE that the $200\times 152$ coefficient matrix of a
basis of the above space has maximal rank. ∎

Using the first $200$ coefficients of ${\rm BB}(q)$ from Appendix A, computed
using buckygen, we can solve a linear system to determine ${\rm BB}(q)$
explicitly. We find that ${\rm BB}(q)$ lies in a much smaller space:

######  Proposition 3.24.

${\rm BB}(q)$ lies in the $47$-dimensional space

$\displaystyle\mathbb{M}=\mathbb{E}_{10}^{1}\oplus\mathbb{E}_{9}^{3}\oplus(\mathbb{E}_{8}^{2}+_{\mathbb{E}_{8}^{1}}\mathbb{E}_{8}^{3})\oplus\mathbb{E}_{7}^{6}\oplus\mathbb{E}_{6}^{6}\oplus\mathbb{E}_{5}^{6}\oplus(\mathbb{E}_{4}^{6}+_{\mathbb{E}_{4}^{2}}\mathbb{E}_{4}^{2,2})\,\oplus$
$\displaystyle(\mathbb{E}_{3}^{6,2}$
$\displaystyle+_{\mathbb{E}_{3}^{3}}\mathbb{E}_{3}^{3,3})\oplus(\mathbb{E}_{2}^{6,2}+_{\mathbb{E}_{2}^{3}}\mathbb{E}_{2}^{3,3})\oplus\mathbb{C}E_{2}(q)\oplus\mathbb{C}E_{2}(q^{5})\oplus_{n\leq
6}\mathbb{C}A(q^{n})\oplus\mathbb{C}$

where $\mathbb{E}_{k}^{N,d}\subset M_{k}^{N,d}$ is the Eisenstein submodule.

Furthermore, all but a small number of coefficients in the expression of ${\rm
BB}(q)$ in the SAGE basis of $\mathbb{M}$ are nonzero, suggesting that this
result is nearly optimal. We now give an explicit formula.

If $\chi$ and $\psi$ are two Dirichlet characters of modulus $L$ and $R$
respectively, with $\psi$ primitive, and $k$ is a positive integer such that
$\chi(-1)\psi(-1)=(-1)^{k}$, the Eisenstein series $E_{k,\chi,\psi}$ is
defined by its Fourier expansion

$E_{k,\chi,\psi}(q)=c_{0}+\sum_{n\geq
1}\left(\sum_{d|n}\chi(n/d)\psi(d)d^{k-1}\right)q^{n}.$

where $c_{0}$ is 0 if $L>1$ and a certain Bernoulli number if $L=1$. If both
characters are the trivial character $\mathbf{1}$, we abbreviate
$E_{k}=E_{k,\mathbf{1},\mathbf{1}}$. If $k\neq 2$, then for any positive
integer $t$, and any $N$ such that $LRt$ divides $N$, the series
$E_{k,\chi,\psi}(q^{t})$ is a modular form of weight $k$ for $\Gamma_{1}(N)$
[26, Thm. 5.8]. If $k=2$, then $E_{2}(q)-tE_{2}(q^{t})$ is a modular form of
weight 2 for $\Gamma_{0}(t)$.

In the following, $\chi_{3}$ denotes the non-trivial Dirichlet character with
modulus 3, and $\mathbf{1}$ the trivial character with modulus 1.

######  Theorem 3.25.

${\rm BB}(q)=$

$\displaystyle\,\,\,\,\,\,\,\tfrac{809}{2^{15}3^{13}5^{2}}E_{10}(q)-\tfrac{1}{2^{16}3^{11}5^{2}}E_{9,\mathbf{1},\chi_{3}}(q)-\tfrac{1}{2^{16}3^{3}5^{2}}E_{9,\chi_{3},\mathbf{1}}(q)+\tfrac{515}{2^{13}3^{11}}E_{8}(q)$
$\displaystyle-\tfrac{1}{2^{6}3^{9}5}E_{8}(q^{2})-\tfrac{1}{2^{11}3^{4}5}E_{8}(q^{3})+\tfrac{47}{2^{14}3^{12}}E_{7,\mathbf{1},\chi_{3}}(q)+\tfrac{1}{2^{8}3^{9}5\cdot
7}E_{7,\mathbf{1},\chi_{3}}(q^{2})$
$\displaystyle-\tfrac{7}{2^{14}3^{3}5}E_{7,\chi_{3},\mathbf{1}}(q)-\tfrac{1}{2^{8}3^{3}5\cdot
7}E_{7,\chi_{3},\mathbf{1}}(q^{2})+\tfrac{2138939}{2^{14}3^{11}5^{2}}E_{6}(q)+\tfrac{3571}{2^{9}3^{8}5}E_{6}(q^{2})$
$\displaystyle-\tfrac{29}{2^{11}3^{6}5}E_{6}(q^{3})+\tfrac{1}{2^{6}3^{3}5}E_{6}(q^{6})+\tfrac{274421}{2^{15}3^{11}5^{2}}E_{5,\mathbf{1},\chi_{3}}(q)-\tfrac{8231}{2^{11}3^{9}5}E_{5,\mathbf{1},\chi_{3}}(q^{2})$
$\displaystyle-\tfrac{2393}{2^{15}3^{2}5^{2}}E_{5,\chi_{3},\mathbf{1}}(q)-\tfrac{203}{2^{11}3^{2}5}E_{5,\chi_{3},\mathbf{1}}(q^{2})+\tfrac{12074417}{2^{13}3^{12}}E_{4}(q)+\tfrac{287281}{2^{10}3^{9}5}E_{4}(q^{2})$
$\displaystyle-\tfrac{2672137}{2^{1}13^{9}5}E_{4}(q^{3})-\tfrac{1}{2^{2}3^{3}}E_{4}(q^{4})-\tfrac{4289}{2^{8}3^{6}}E_{4}(q^{6})+\tfrac{3343037}{2^{14}3^{11}}E_{3,\mathbf{1},\chi_{3}}(q)$
$\displaystyle+\tfrac{227779}{2^{12}3^{8}5}E_{3,\mathbf{1},\chi_{3}}(q^{2})+\tfrac{1}{2^{3}3^{3}}E_{3,\mathbf{1},\chi_{3}}(q^{3})+\tfrac{1}{2^{5}3^{2}}E_{3,\mathbf{1},\chi_{3}}(q^{4})-\tfrac{33103}{2^{14}3^{3}5}E_{3,\chi_{3},\mathbf{1}}(q)$
$\displaystyle-\tfrac{713}{2^{12}3\cdot
5}E_{3,\chi_{3},\mathbf{1}}(q^{2})+\tfrac{1}{2^{3}3}E_{3,\chi_{3},\mathbf{1}}(q^{3})-\tfrac{1}{2^{5}3}E_{3,\chi_{3},\mathbf{1}}(q^{4})+\tfrac{4314057521}{2^{15}3^{11}5^{2}}E_{2}(q)$
$\displaystyle-\tfrac{59136661}{2^{14}3^{9}5}E_{2}(q^{2})-\tfrac{32541151}{2^{11}3^{10}5}E_{2}(q^{3})+\tfrac{1}{2^{5}3}E_{2}(q^{4})+\tfrac{2}{3\cdot
5}E_{2}(q^{5})+\tfrac{5023687}{2^{10}3^{8}5}E_{2}(q^{6})$
$\displaystyle+\tfrac{1}{2^{6}}E_{2}(q^{9})+\tfrac{2}{3^{3}}E_{2}(q^{12})-\tfrac{24019585289}{2^{15}3^{13}5^{2}}A(q)-\tfrac{3212143}{2^{11}3^{9}5}A(q^{2})-\tfrac{1}{2^{2}3^{4}}A(q^{3})$
$\displaystyle-\tfrac{7}{2^{7}3^{3}}A(q^{4})-\tfrac{2}{3\cdot
5}A(q^{5})-\tfrac{1}{2\cdot
3^{2}}A(q^{6})+\tfrac{3608212332449}{2^{18}3^{14}5\cdot 11}$

where $A=E_{1,\mathbf{1},\chi_{3}}$.

######  Example 3.26.

If $n$ is congruent to $1$ mod $3$, and not divisible by $2$ or $5$, the
number $\mathrm{BB}_{n}$ of fullerenes with $2n$ carbon atoms is
$\sum_{d|n}\chi_{3}(d)p(\chi_{3}(d)d)$, where $p(d)$ is the polynomial

$\displaystyle p(d)=\frac{1}{2^{15}3^{13}5^{2}}\big{(}809d^{9}$
$\displaystyle-29529d^{8}+463500d^{7}-4126380d^{6}$ $\displaystyle+3$
$\displaystyle 8500902d^{5}-421442982d^{4}+3622325100d^{3}$
$\displaystyle-18042623820d^{2}+38826517689d-24019585289\big{)}.$

In particular, if $n$ is prime and congruent to $1$ mod $3$, then
$\mathrm{BB}_{n}=p(n)$.

######  Example 3.27.

There are $1203397779055806181762759$ fullerenes with $10000$ carbon atoms.
This trails the leading term $\frac{809}{2^{15}3^{13}5^{2}}\sigma_{9}(5000)$
by about $0.73\%$.

######  Remark 3.28.

The final answer is a linear combination of Eisenstein series because of the
Siegel-Weil formula and the fact that each curvature stratum lattice is unique
in its genus. Roughly, this follows from the fact that the Eisenstein root
lattices embed uniquely up to isometry into $\Lambda$.

######  Remark 3.29.

One strange occurrence is the non-appearance of the generating function
$A(q^{10})$ for the icosahedral bistratum. Essentially, it means that all the
various inclusions and exclusions of this bistratum end up cancelling
perfectly. We do not know why this is the case.

##  Appendix A buckygen counts (by Jan Goedgebeur)

In [4] an algorithm is described to generate all pairwise non-isomorphic
fullerenes up to a given order $n$. An efficient implementation of this
algorithm was made in the program called buckygen (of which the source code is
available at [6]). This program was already used in [4] to generate all

Table 2. Counts $\mathrm{BB}_{n}$ of oriented fullerenes with $2n$ vertices $2n$ | $\mathrm{BB}_{n}$ | $2n$ | $\mathrm{BB}_{n}$ | $2n$ | $\mathrm{BB}_{n}$ | $2n$ | $\mathrm{BB}_{n}$
---|---|---|---|---|---|---|---
20 | 1 | 116 | 2 411 814 | 212 | 739 119 987 | 308 | 23 803 804 599
22 | 0 | 118 | 2 814 401 | 214 | 803 025 838 | 310 | 25 177 747 311
24 | 1 | 120 | 3 345 147 | 216 | 880 381 442 | 312 | 26 820 388 881
26 | 1 | 122 | 3 882 755 | 218 | 954 791 893 | 314 | 28 342 433 015
28 | 3 | 124 | 4 588 131 | 220 | 1 045 149 256 | 316 | 30 170 072 140
30 | 3 | 126 | 5 297 876 | 222 | 1 131 745 304 | 318 | 31 860 956 489
32 | 10 | 128 | 6 224 194 | 224 | 1 236 558 914 | 320 | 33 883 714 816
34 | 9 | 130 | 7 157 006 | 226 | 1 337 268 127 | 322 | 35 760 178 173
36 | 23 | 132 | 8 359 652 | 228 | 1 458 771 168 | 324 | 38 003 821 487
38 | 30 | 134 | 9 570 462 | 230 | 1 575 048 577 | 326 | 40 074 384 246
40 | 66 | 136 | 11 128 035 | 232 | 1 715 797 098 | 328 | 42 560 801 885
42 | 80 | 138 | 12 683 755 | 234 | 1 850 016 768 | 330 | 44 852 179 762
44 | 162 | 140 | 14 676 481 | 236 | 2 011 965 672 | 332 | 47 592 925 209
46 | 209 | 142 | 16 671 248 | 238 | 2 166 827 672 | 334 | 50 126 102 754
48 | 374 | 144 | 19 201 153 | 240 | 2 353 180 355 | 336 | 53 155 439 383
50 | 507 | 146 | 21 728 036 | 242 | 2 530 571 274 | 338 | 55 939 718 941
52 | 835 | 148 | 24 930 330 | 244 | 2 744 801 058 | 340 | 59 284 163 396
54 | 1 113 | 150 | 28 109 625 | 246 | 2 948 134 826 | 342 | 62 354 562 777
56 | 1 778 | 152 | 32 122 355 | 248 | 3 192 869 772 | 344 | 66 028 033 946
58 | 2 344 | 154 | 36 112 223 | 250 | 3 425 774 760 | 346 | 69 410 105 709
60 | 3 532 | 156 | 41 107 620 | 252 | 3 705 433 966 | 348 | 73 456 137 608
62 | 4 670 | 158 | 46 064 096 | 254 | 3 970 400 266 | 350 | 77 160 823 316
64 | 6 796 | 160 | 52 272 782 | 256 | 4 289 774 460 | 352 | 81 612 318 561
66 | 8 825 | 162 | 58 393 248 | 258 | 4 591 482 273 | 354 | 85 683 960 523
68 | 12 501 | 164 | 66 032 535 | 260 | 4 953 928 565 | 356 | 90 556 797 909
70 | 16 091 | 166 | 73 582 782 | 262 | 5 297 277 909 | 358 | 95 026 883 784
72 | 22 142 | 168 | 82 940 953 | 264 | 5 708 945 811 | 360 | 100 377 559 914
74 | 28 232 | 170 | 92 160 881 | 266 | 6 097 098 730 | 362 | 105 257 194 470
76 | 38 016 | 172 | 103 602 394 | 268 | 6 564 284 155 | 364 | 111 124 529 629
78 | 47 868 | 174 | 114 816 693 | 270 | 7 003 726 743 | 366 | 116 472 010 663
80 | 63 416 | 176 | 128 686 912 | 272 | 7 530 784 730 | 368 | 122 874 832 878
82 | 79 023 | 178 | 142 308 166 | 274 | 8 027 877 473 | 370 | 128 726 803 405
84 | 102 684 | 180 | 159 057 604 | 276 | 8 623 164 190 | 372 | 135 735 763 019
86 | 126 973 | 182 | 175 455 386 | 278 | 9 181 941 021 | 374 | 142 104 866 311
88 | 162 793 | 184 | 195 657 995 | 280 | 9 853 806 446 | 376 | 149 768 451 649
90 | 199 128 | 186 | 215 335 951 | 282 | 10 482 937 182 | 378 | 156 727 483 454
92 | 252 082 | 188 | 239 498 752 | 284 | 11 236 734 954 | 380 | 165 065 400 161
94 | 306 061 | 190 | 263 096 297 | 286 | 11 944 678 343 | 382 | 172 658 723 068
96 | 382 627 | 192 | 291 923 618 | 288 | 12 791 770 724 | 384 | 181 761 605 787
98 | 461 020 | 194 | 319 971 240 | 290 | 13 583 361 145 | 386 | 190 001 957 648
100 | 570 603 | 196 | 354 321 904 | 292 | 14 534 386 155 | 388 | 199 925 650 318
102 | 682 017 | 198 | 387 597 455 | 294 | 15 421 369 613 | 390 | 208 906 495 800
104 | 836 457 | 200 | 428 220 000 | 296 | 16 483 227 350 | 392 | 219 673 875 162
106 | 993 461 | 202 | 467 658 270 | 298 | 17 476 267 675 | 394 | 229 445 258 636
108 | 1 206 782 | 204 | 515 596 902 | 300 | 18 663 927 061 | 396 | 241 169 817 560
110 | 1 424 663 | 206 | 561 974 100 | 302 | 19 768 989 050 | 398 | 251 745 893 960
112 | 1 718 034 | 208 | 618 503 629 | 304 | 21 096 191 945 | 400 | 264 495 159 034
114 | 2 014 713 | 210 | 672 960 919 | 306 | 22 328 857 779 |  |

pairwise non-isomorphic (unoriented) fullerenes up to 400 vertices. These
counts were independently confirmed by the generator fullgen of Brinkmann and
Dress [3] up to 380 vertices.

By default, buckygen outputs or counts one graph from each isomorphism class.
However, there is also a built-in option to write one member of each
orientation-preserving isomorphism class (by using the flag `-o`). In this
case, the output graphs are tested for the presence of an orientation-
reversing automorphism. If there is none, the mirror image of the graph is
output as well. We used this option to generate all oriented fullerenes (i.e.
considering enantiomorphic fullerenes as distinct) up to 400 vertices. The
computation took approximately 3 CPU years and was performed on the
supercomputer of the VSC (Flemish Supercomputer Center). The counts are found
in Table 2. Previously these counts were determined up to 98 vertices (cf.
sequence A057210 in the On-Line Encyclopedia of Integer Sequences [25]).

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