arXiv:1906.06068v1 [math.GR] 14 Jun 2019GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES
OF QUANTUM COMPUTING
MICHEL PLANAT†, RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡ AND KLEE IRWIN‡
Abstract. Let H be a non trivial subgroup of index d of a free group
G and N the normal closure of H in G. The coset organization in a sub-
group H of G provides a group P of permutation gates whose common
eigenstates are either stabilizer states of the Pauli group or magic states
for universal quantum computing. A subset of magic states consists
of MIC states associated to minimal informationally complete measure-
ments. It is shown that, in most cases, the existence of a MIC state
entails that the two conditions (i) N = G and (ii) no geometry (a triple
of cosets cannot produce equal pairwise stabilizer subgroups), or that
these conditions are both not satisfied. Our claim is verified by defin-
ing the low dimensional MIC states from subgroups of the fundamental
group G = π1(M) of some manifolds encountered in our recent papers,
e.g. the 3-manifolds attached to the trefoil knot and the figure-eight
knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to
the aforementioned rule are classified in terms of geometric contextuality
(which occurs when cosets on a line of the geometry do not all mutually
commute).
MSC codes: 81P68, 51E12, 57M05, 81P50, 57M25, 57R65, 14H30
1. Introduction
Interpreting quantum theory is a long standing effort and not a single
approach can exhaust all facets of this fascinating subject. Quantum infor-
mation owes much to the concept of a (generalized) Pauli group for under-
standing quantum observables, their commutation, entanglement, contextu-
ality and many other aspects, e.g. quantum computing. Quite recently, it
has been shown that quantum commutation relies on some finite geometries
such as generalized polygons and polar spaces [1]. Finite geometries connect
to the classification of simple groups as understood by prominent researchers
as Jacques Tits, Cohen Thas and many others [2, 3].
In the Atlas of finite group representations [4], one starts with a free
group G with relations, then the finite group under investigation P is the
permutation representation of the cosets of a subgroup of finite index d of G
(obtained thanks to the Todd-Coxeter algorithm). As a way of illustrating
this topic, one can refer to [3, Table 3] to observe that a certain subgroup of
index 15 of the symplectic group S′4(2) corresponds to the 2QB (two-qubit)
commutation of the 15 observables in terms of the generalized quadrangle
of order two, denoted GQ(2, 2) (alias the doily). For 3QB, a subgroup of
index 63 in the symplectic group S6(2) does the job and the commutation
relies on the symplectic polar spaceW5(2) [3, Table 7]. An alternative way to
approach 3QB commutation is in terms of the generalized hexagon GH(2, 2)
1
2MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
(or its dual) which occurs from a subgroup of index 63 in the unitary group
U3(3) [3, Table 8]. Similar geometries can be defined for multiple qudits
(instead of qubits).
The straightforward relationship of quantum commutation to the appro-
priate symmetries and finite groups was made possible thanks to techniques
developed by the first author (and coauthors) that we briefly summarize.
This will be also useful at a later stage of the paper with the topic of magic
state quantum computing.
The rest of this introduction recalls how the permutation group organizing
the cosets leads to the finite geometries of quantum commutation (in Sec.
1.1) and how it allows the computation of magic states of universal quantum
commutation (in Sec. 1.2). In this paper, it is shown that magic states
themselves may be classified according to their coset geometry with two
simple axioms (in Sec. 1.3).
1.1. Finite geometries from cosets [3, 6, 7]. One needs to define the
rank r of a permutation group P . First it is expected that P acts faithfully
and transitively on the set Ω = {1, 2, · · · , n} as a subgroup of the symmetric
group Sn. The action of P on a pair of distinct elements of Ω is defined as
(α, β)p = (αp, βp), p ∈ P , α
6= β. The orbits of P on Ω×Ω are called orbitals,
and the number of orbits is called the rank r of P on Ω. The rank of P is at
least two, and the two-transitive groups identify to the rank two permutation
groups. Next, selecting a pair (α, β) ∈ Ω × Ω, α
6= β, one introduces the
two-point stabilizer subgroup P(α,β) = {p ∈ P |(α, β)
p = (α, β)}. There
exist 1 < m ≤ r such non-isomorphic (two-point stabilizer) subgroups Sm of
P . Selecting one with α
6= β, one defines a point/line incidence geometry G
whose points are the elements of Ω and whose lines are defined by the subsets
of Ω sharing the same two-point stabilizer subgroup. Thus, two lines of G
are distinguished by their (isomorphic) stabilizers acting on distinct subsets
of Ω. A non-trivial geometry arises from P as soon as the rank of the
representation P of P is r > 2, and simultaneously, the number of non
isomorphic two-point stabilizers of P is m > 2. Further, G is said to be
contextual (shows geometrical contextuality) if at least one of its lines/edges
corresponds to a set/pair of vertices encoded by non-commuting cosets [7].
Figure 1 illustrates the application of the two-point stabilizer subgroup
approach just described for the index 15 subgroup of the symplectic group
is S′4(2) = A6 whose finite representation is
H =
〈
a, b|a2 = b4 = (ab)5 = (ab2)5 = 1
〉
. The finite geometry organizing
the coset representatives is the generalized quadrangle GQ(2, 2). The other
reresentation is in terms of the two-qubit Pauli operators, as first found in
[1, 8]. It is easy to check that all lines not passing through the coset e con-
tains some mutually not commuting cosets so that the GQ(2, 2) geometry is
contextual. The embedded (3×3)-grid shown in bold (the so-called Mermin
square) allows a 2QB proof of Kochen-Specker theorem [5].
1.2. Magic states in quantum computing. Now we recall our recent
work about the relation of coset theory to the magic states of universal
quantum computing. Bravyi & Kitaev introduced the principle of ‘magic
state distillation’ [9]: universal quantum computation, the possibility of
GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES OF QUANTUM COMPUTING3
Figure 1. The generalized quadrangle of order two
GQ(2, 2). The picture provides a representation in terms of
the fifteen 2QB observables that are commuting by triples:
the lines of the geometry. Bold lines are for an embedded
3×3 grid (also called Mermin square) that is a basic model of
Kochen-Specker theorem (e.g. [3, Fig.1] or [5]). The second
representation is in terms of the cosets of the permutation
group arising from the index 15 subgroup of G ∼= A6 (the
6-letter alternating group).
getting an arbitrary quantum gate, may be realized thanks to stabilizer op-
erations (Clifford group unitaries, preparations and measurements) and an
appropriate single qubit non-stabilizer state, called a ‘magic state’. Then,
irrespective of the dimension of the Hilbert space where the quantum states
live, a non-stabilizer pure state was called a magic state [10]. An improve-
ment of this concept was carried out in [11] showing that a magic state
could be at the same time a fiducial state for the construction of a minimal
informationally complete positive operator-valued measure, or MIC, under
the action on it of the Pauli group of the corresponding dimension. Thus
UQC in this view happens to be relevant both to magic states and to MICs.
In [11], a d-dimensional magic state is obtained from the permutation group
that organizes the cosets of a subgroup H of index d of a two-generator
free group G. This is due to the fact that a permutation may be realized
as a permutation matrix/gate and that mutually commuting matrices share
eigenstates - they are either of the stabilizer type (as elements of the Pauli
group) or of the magic type. It is enough to keep magic states that are
4MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
simultaneously fiducial states for a MIC because the other magic states may
loose the information carried during the computation. A catalog of the
magic states relevant to UQC and MICs can be obtained by selecting G as
the two-letter representation of the modular group Γ = PSL(2,Z) [12]. The
next step, developed in [13, 14], is to relate the choice of the starting group
G to three-dimensional topology. More precisely, G is taken as the funda-
mental group π1(S
3 \L) of a 3-manifold M3 defined as the complement of a
knot or link L in the 3-sphere S3. A branched covering of degree d over the
selected M3 has a fundamental group corresponding to a subgroup of index
d of π1(M
3) and may be identified as a sub-manifold of M3, the one leading
to a MIC is a model of UQC. In the specific case of Γ, the knot involved is
the left-handed trefoil knot T1 = 3
1, as shown in [12] and [13, Sec. 2].
1.3. Coset geometry of magic states. The goal of the paper is to classify
the magic states according to the coset geometry where they arise. We start
from a non trivial subgroup H of index d of a free group G and we denote
N the normal closure of H in G. As above, the coset organization in a
subgroup H of G provides a group P of permutation gates whose common
eigenstates are either stabilizer states of the Pauli group or magic states for
universal quantum computing. A subset of magic states consists of MIC
states associated to minimal informationally complete measurements.
It is shown in the paper that, in many cases, the existence of a MIC
state entails that the two conditions (i) N = G and (ii) no geometry (a
triple of cosets cannot produce equal pairwise stabilizer subgroups), or that
these conditions are both not satisfied. Our claim is verified by defining
the low dimensional MIC states from subgroups of the fundamental group
G = π1(M) of manifolds encountered in our recent papers, e.g.
the 3-
manifolds attached to the trefoil knot and the figure-eight knot, and the
4-manifolds defined by 0-surgery of them.
Exceptions to the aforementioned rule are classified in terms of geometric
contextuality (which occurs when cosets on a line of the geometry do not
all mutually commute).
In section 2, one deals with the case of MIC states obtained from the sub-
groups of the fundamental group of Figure-of-Eight knot hyperbolic manifold
and its 0-surgery. In section 3, the MIC states produced with the trefoil knot
manifold and its 0-surgery are investigated.
2. MIC states pertaining to the Figure-of-Eight knot and its
0-surgery
We first investigate the relation of MIC states to the group geometrical
axioms (i)-(ii) (or their negation) in the context of the Figure-of-Eight knot
K4a1 (in Sec. 2.2) and its 0-surgery (in Sec. 2.1). The fundamental group
of the complement of K4a1 in the 3-sphere G = π1(S
3 \ K4a1), and its
connection to MICs, is first studied in [13, Table 2], below are new results
and corrections.
2.1. Group geometrical axioms applied to the fundamental group
π1(Y ). The manifold Y defined by 0-surgery on the knot K4a1 is of special
GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES OF QUANTUM COMPUTING5
interest as shown in [14, Sec 2] and references therein. The number of
subgroups of index d of the fundamental group π1(Y ) is as follows
ηd[π1(Y )] = [1, 1, 1,2, 2, 5, 1, 2,2, 4, 3, 17, 1, 1, 2, 3, 1, 6, 3, 6, 1, 3, 1, 43, . . .],
where a bold number means that a MIC exists at the corresponding index.
In Table 1, one summarizes the check of our axioms (i) and (ii) applied
to π1(Y ). A triangle ∆ means that a geometry does exist (corresponding
to at least a triple of cosets with equal pairwise stabilizer subgroups), thus
with (ii) is violated. According to our theory, for a MIC to exist, we should
have (i) and (ii) satisfied, or both of them violated. The former case occurs
for d = 9, 11 and 19. The latter case occurs for d = 6 where the geometry
is that of the octahedron (with the 3-partite graph K(2, 2, 2)) and d = 20,
where the geometry is encoded by the the complement of the line graph of
the bipartite K(4, 5). In all of these five cases, a pp-valued MIC does exist.
For dimension 4, the bold triangle points out a violation since (i) is true
and (ii) is false while the 2QB-MIC exists. In this case the geometry is the
tetrahedron (with complete graph K4) but not all cosets on a line/triangle
are mutually commuting, a symptom of geometric contextuality, as shown
in Fig. 2.
Figure 2. The contextual geometry associated to the 2QB-
MIC and permutation group P = A4 in Table 1. The
line/triangle {a, ab, ab−1} is not made of mutually commut-
ing cosets, thus geometric contextuality occurs.
2.2. Group geometrical axioms applied to the fundamental group
π1(S
3 \K4a1). The submanifolds obtained from the subgroups of index
d of the fundamental group π1(S
3 \ K4a1) for the Figure-of-Eight knot
complement are given in Table 2 (column 3), as identified in SnapPy [15]
(this corrects a few mistakes of [13, Table 2]).
As for the subsection above, when axioms (i) and (ii) are simultaneously
satisfied (or both are not satisfied), a MIC is created. Otherwise no MIC
exist in the corresponding dimension, as expected.
There are three exceptions where (i) is true and a geometry does exist
(when (ii) fails to be satisfied). This first occurs in dimension 4 with a
2QB MIC arising from the 3-manifold otet0800002 , in this case geometric
6MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
d P
(i) pp
geometry
4 A4
yes
2
2QB MIC, ∆
5
10
yes
∆
6 A4
no
2
6-dit MIC, K(2, 2, 2)
9
(36, 9) ∼= 32 ⋊ 4
yes
2
2QT MIC
11 (55, 1) = 11⋊ 5, (×2)
yes
3
11-dit MIC
16 (48, 3) ∼= 4⋊A4
yes
∆
19 (171, 3) ∼= 19 ⋊ 9
yes
3
19-dit MIC
20 (120, 39) ∼= 4⋊ (5⋊ (6, 2)) yes
L(K(4, 5))
Table 1. Table of subgroups of the fundamental group
π1[S
3 \ K4a1(0, 1)] [with K4a1(0, 1) the 0-surgery over the
Figure-of-Eight knot]. The permutation group P organiz-
ing the cosets in column 2. If (i) is true, unless otherwise
specified, the graph of cosets leading to a MIC is that of the
d-simplex [ and/or the condition (ii) is true: no geometry].
The symbol ∆ means that (ii) fails to be satisfied. When
there exists a MIC with (i) true and (ii) false, the geometry
is shown in bold characters (here this occurs in dimension 4,
see Fig. 2). If it exists, the MIC is pp-valued as given in col-
umn 4. In addition, K(2, 2, 2) is the binary tripartite graph
(alias the octahedron) and L(K(4, 5)) means the complement
of the line graph of the bipartite graph K(4, 5).
contextuality occurs in the cosets as in Fig. 2 of the previous subsection.
Then, it occurs in dimension 7 (corresponding to 3-manifolds otet1400002 and
otet1400035) when the geometry of cosets is that of the Fano plane shown in
Fig. 3a. Finally it occurs in dimension 10 when the geometry of cosets is
that of a [103] configuration shown in Fig. 3b [16, p 74].
In addition to the latter cases, false detection of a MIC may occur (this
is denoted ‘fd’) in dimension 8 as shown in Table 2.
GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES OF QUANTUM COMPUTING7
Figure 3. Contextual geometries associated to ‘(i) true and
(ii) false’ for the MICs of the Figure-of-Eight knot K4a1
listed in Table 1: (a) the Fano plane related to the manifold
otet1400002 at index 7, (b) the configuration [103] at index
10. The bold lines are for cosets that are not all mutually
commuting. Each line corresponds to pair of cosets with the
same stabilizer subgroup isomorphic to Z22.
3. MIC states pertaining to the Trefoil knot and its 0-surgery
We now investigate the relation of MIC states to the group geometrical axioms
(i)-(ii) (or their negation) in the context of the trefoil knot 31 (in Sec. 3.2) and
its 0-sugery (in Sec. 3.1). The fundamental group of the complement of 31 in the
3-sphere G = π1(S
3 \ 31), and its connection to MICs, is studied in [13, Table 1]
and below.
3.1. Group geometrical axioms applied to the fundamental group π1(Ẽ8).
The manifoldẼ8 is defined by 0-surgery on the trefoil knot 31 and is of special
interest as shown in [14, Sec 3] and references therein. The number of subgroups
of index d of the fundamental group π1(Y ) is as follows
ηd[Ẽ8] = [1, 1,2,2, 1, 5,3, 2, 4, 1, 1, 12,3, 3,4, 3, 1, 17,3, 2, 8, 1, 1, 27, 2, . . .]
where a bold number means that a MIC exists at the corresponding index.
Such cases are summarized in Table 3. As expected this occurs when the axioms
(i) and (ii) are both true, or are both false. The latter case occurs at index 6 with
geometry of the octahedron [and graph K(2, 2, 2)] and at index 15 with a geometry
of graph K(5, 5, 5).
Exceptions to the rules are when a MIC exists with (i) true but not (ii). This
occurs in dimension 3 (for the Hesse SIC) since the free group has a single gen-
erator (a trivial case), at index 4 (for the 2QB MIC) with a contextual geometry
as in Fig. 2 and at index 21 with a contextual geometry (not shown) of graph
K(3, 3, 3, 3, 3, 3, 3).
3.2. Group geometrical axioms applied to the fundamental group π1(S
3\
31). The characteristics of submanifolds obtained from the subgroups of index d
of the fundamental group π1(S
3 \ 31) for the trefoil knot complement are given
8MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
d
ty
M3 (or P )
cp
(i) pp
geometry
2
cyc
otet0400002, m206
1
no
3
cyc
otet0600003, s961
1
no
4
irr
otet0800002, L10n46, t12840
2
yes 2
2QB MIC, ∆
cyc
otet0800007, t12839
1
no
5
cyc
otet1000019
1
no
irr
otet1000006, L8a20
3
yes
∆
irr (×2)
otet1000026
2
yes 1
5-dit MIC
6
cyc
otet1200013
1
no
irr
otet1200039
1
no
irr (×2)
otet1200038
1
yes 10
6-dit MIC
irr
otet1200041
2
no
irr (×2)
otet1200017
2
no
irr (×4)
otet1200000
2
yes 2
6-dit MIC
7
cyc
otet1400019
1
no
irr (×4)
otet1400002, L14n55217
3
yes 2
7-dit MIC,∆ : Fano
irr (×4)
otet1400035
1
yes 2
7-dit MIC, ∆ : Fano
8
cyc
otet1600026
1
no
irr (×2)
otet1600035
1
no
irr (×2)
otet1600079
2
yes
fd
irr (×2)
otet1600016
2
yes
fd
irr
otet1600092
2
no
irr
otet1600091
2
yes
16-cell
irr
otet1600013, L14n17678
2
no
9
(36, 9) ∼= 32 ⋊ 4
yes 2
2QT MIC
(×2)
(504, 156) = PSL(2, 8)
yes 3
2QT MIC
(×2)
(216, 153) ∼= 32 ⋊ (24, 3)
yes 2
2QT MIC
10 (×6)
(160, 234) ∼= 24 ⋊ 10
yes 5
10-dit MIC
(×2)
(120, 34) = S5
yes 4
10-dit MIC, ∆ : [103]
(×2)
(120, 34) = S5
no 7
10-dit MIC, 5-ortho
(360, 118) = A6
yes 5
10-dit MIC
Table 2. Table of 3-manifolds M3 found from subgroups of finite
index d of the fundamental group π1(S
3 \K4a1) (alias the d-fold cover-
ings over the Figure-of-Eight knot 3-manifold). The covering type ‘ty’
in column 2, the manifold identification ‘M3’ in column 3 and the num-
ber of cusps ‘cp’ in column 4 are from SnapPy [15]. For d = 9 and 10,
SnapPy does not provide results so that we only identify the permuta-
tion group P =SmallGroup(o, k) (abbreviated as (o, k)), where o is the
order and k is the k-th group of order o in the standard notation (that
is used in Magma). If it exists, the MIC is ‘pp’-valued. If (i) is true,
unless otherwise specified, the graph of cosets leading to a MIC is that
of the d-simplex [and/or the condition (ii) is true: no geometry]. The
symbol ∆ means that (ii) fails to be satisfied. When there exists a MIC
with (i) true and (ii) false, the geometry is shown in bold characters.
The symbol ‘fd’ means a false detection of a MIC when (i) and (ii) are
satisfied simultaneously while a MIC does not exist. The abbreviations
‘Fano’, ‘d-ortho’ and ‘[103]’ are for the Fano plane, the d-orthoplex and
the corresponding geometric configuration.
in Table 4 by using SnapPy [15] and Sage [17] for identifying the corresponding
subgroup of the modular group Γ [12] (this improves [13, Table 1]).
GROUP GEOMETRICAL AXIOMS FOR MAGIC STATES OF QUANTUM COMPUTING9
d P
(i)
pp
geometry
3
6
yes
1
Hesse SIC, ∆
4 A4
yes
2
2QB MIC, ∆
6 A4
no
2
6-dit MIC, K(2, 2, 2)
7
(42, 1) ∼= 7⋊ (6, 2)
yes
2
7-dit MIC
9
(54, 5) ∼= 32 ⋊ (18, 3), (×2) yes
K(3, 3, 3)
12 (72, 44) ∼= 22 ⋊ (18, 3)
yes
L(K(3, 4))
13 (78, 1) ∼= 13⋊ (6, 2), (×2)
yes
4
13-dit MIC
15 (150, 6) ∼= 52 ⋊ (6, 2), (×2)
no
6
15-dit MIC, K(5, 5, 5)
16 (96, 72) ∼= 23 ⋊A4
yes
K(4, 4, 4, 4)
19 (114, 1) ∼= 19⋊ (6, 2)
yes
3
19-dit MIC
21 (126, 9) ∼= 7⋊ (18, 3), (×2) yes
5
21-dit MIC, ∆: K(3,3,3,3,3,3,3)
Table 3. Table of subgroups of the fundamental group
π1[S
3 \ 31(0, 1)] [with 31(0, 1) the 0-surgery over the trefoil
knot] when the condition (i) is satisfied or when a MIC is
missed. See the captions of Table 1 and 2 for the meaning of
abbreviations.
As for the above sections, when axioms (i) and (ii) are simultaneously satisfied
(or both are not satisfied), a MIC is created. Otherwise no MIC exist in the
corresponding dimension, as one should expect.
There are a few exceptions where (i) is true and a geometry does exist (when
(ii) fails to be satisfied). This first occurs in dimension 3 for the Hesse SIC where
the free subgroup is trivial with a single generator. The next exceptions are for the
6-dit MIC related to the permutation group S4 with the contextual geometry of
the octahedron shown in Fig. 4a, in dimension 9 for the 2QT MIC related to the
permutation group 33 ⋊ S4 with a contextual geometry consisting of three disjoint
lines, and in dimension 10 for a 10-dit MIC related to the permutation group A5 and
the contextual geometry of the so-called Mermin pentagram. The latter geometry
is known to allow a 3QB proof of the Kochen-Specker theorem [5].
4. Conclusion
Previous work about the relationship between quantum commutation and
coset-generated finite geometries has been expanded here by establishing a
connection between coset-generated magic states and coset-generated finite
geometries. The magic states under question are those leading to MICs
(with minimal complete quantum information in them). We found that,
given an appropriate free group G, two axioms (i): the normal closure N of
the subgroup of G generating the MIC is G itself and (ii): no coset-geometry
should exist, or the negation of both axioms (i) and (ii), are almost enough
to classify the MIC states. The few exceptions rely on configurations that
admit geometric contextuality. We restricted the application of the theory
to the fundamental group of the 3-manifolds defined from the Figure-of-
Eight knot (an hyperbolic manifold) and from the trefoil knot, and to 4-
manifolds Y andẼ8 obtained by 0-surgery on them. It is of importance
to improve of knowledge of the magic states due to their application to
quantum computing and we intend to pursue this research in future work.
10MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 4. Contextual geometries associated to ‘(i) true and
(ii) false’ for the MICs of the trefoil knot 31 listed in Table
2: (a) the octahedron, related to the subgroup Γ0(4) of Γ
at index 6 , (b) three disjoint lines K33 at index 9, (c) The
Mermin’s pentagram at index 10. The bold lines are for
cosets that are not all mutually commuting.
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[12] M. Planat, The Poincaré half-plane for informationally complete POVMs, Entropy
20 16 (2018).
[13] M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, Universal quantum computing
and three-manifolds, Universal quantum computing and three-manifolds Symmetry
10 773 (2018).
[14] M. Planat, R. Aschheim, M. M. Amaral and K. Irwin, Quantum computing, Seifert
surfaces and singular fibers, Quantum Reports 1 12-22 (2019).
[15] M. Culler, N. M. Dunfield, M. Goerner,
and J. R. Weeks, SnapPy, a
computer program for studying the geometry and topology of 3-manifolds,
http://snappy.computop.org.
[16] B. Grünbaum, Configurations of points and lines (Graduate studies in mathematics,
vol. 103, AMS, Providence, Rhode Island, 2009).
[17] William A. Stein et al. Sage Mathematics Software (Version 8.6), The Sage Develop-
ment Team, 2019, http://www.sagemath.org.
† Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR
6174, 15 B Avenue des Montboucons, F-25044 Besançon, France.
E-mail address: michel.planat@femto-st.fr
‡ Quantum Gravity Research, Los Angeles, CA 90290, USA
E-mail address: raymond@QuantumGravityResearch.org
E-mail address: Klee@quantumgravityresearch.org
E-mail address: Marcelo@quantumgravityresearch.org
12MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
d
ty
cp
P
(i) pp
type in Γ geometry
2
cyc
1
(2,1) ≡ 2
no
3
cyc
1
(3,1) ≡ 3
no
irr
2
(6,1) ≡ 6
yes 1
Γ0(2) Hesse SIC, ∆, L7n1
4
cyc
1
(4,1) ≡ 4
no
irr
2
(12, 3) = A4
yes 2
Γ0(3)
2QB MIC, ∆, L6a3
irr
1
(24, 12) = S4
yes 2
4A0 2QB MIC
5
cyc
2
(5,1) ≡ 5
no
irr
3
(60, 5) = A5
yes 1
5A0 5-dit MIC
6
reg
3
(6,1) ≡ 6
no 2
Γ(2)
6-dit MIC, 63
3
[14]
cyc
3
(6,2) = 3×2
no
Γ′
irr
2
A4
no 2
3C0
6-dit MIC, K(2,2,2)
irr
1
(24, 13) = 3⋊ 8
no
6B0
irr
1
(18, 3) ∼= 32 ⋊ 2
no
6A0
irr
3
S4
yes 2
Γ0(4)
6-dit MIC, ∆ : octa
irr
2
A5
yes 2
Γ0(5) 6-dit MIC
irr
2
S4
yes 2
4C0
6-dit MIC, ∆ : octa
7
cyc
1
(7,1)≡ 7
no
irr (×2)
2
(42, 1) ∼= 7 ⋊ (6, 2)
yes 2
NC(0, 6, 1, 1, [1161]) 7-dit MIC
irr (×2)
1
(168, 42) = PSL(2, 7)
yes 2
7A0 7-dit MIC
irr (×2)
2
S7 (order 5040)
yes
NC(0, 10, 1, 1, [2151]) ? 7-dit MIC
8
cyc
1
(8, 1) ≡ 8
no
irr
2
(24,13)
no
6C0
irr
2
S4
no
4D0
irr (×2)
2
(24, 3) ∼= 2.A4
yes
16-cell
irr
2
PSL(2, 7)
yes
Γ0(7)
fd
irr (×2)
1
SL(2, 7)
yes
NC(0, 8, 2, 2, [81])
fd
irr (×2)
2
(48, 29) ∼= 2.(24, 3)
yes
8A0 16-cell
9
(9, 1) ≡ 9
no
2
(18,3)
no 7
6D0
9-dit MIC, K(3,3,3)
2
(54, 5) ∼= 32 ⋊ (18, 3)
no 7
NC(0, 6, 3, 0, [3161])
9-dit MIC, K(3,3,3)
1
(324, 160) ∼= 33 ⋊ A4
no
9A0 K(3,3,3), K33
3
(54,5)
yes 7
NC(0, 6, 1, 0, [112161]) 9-dit MIC
(×3)
(162, 10) ∼= 32 ⋊ 6
yes
K(3,3,3)
(×2)
1
(504, 156) = PSL(2, 8)
yes 3
NC(1, 9, 1, 0, [91]) 2QT MIC
(×2)
2
(432, 734) ∼= 32 ⋊ (48, 29) yes 2
NC(0, 8, 3, 0, [8111]) 2QT MIC
3
(648, 703) ∼= 33 ⋊ S4
yes 2
NC(0, 12, 1, 0, [213141])
2QT MIC, ∆ : K3
3
10
1
(120, 35) ∼= 2 ⋊ A5
no
10A0 5-ortho
2
A5
yes 5
5C0 10-dit MIC, ∆: MP
(×2)
1
(720, 764) ∼= A6 ⋊ 2
yes 9
NC(0, 10, 0, 4, [101]) 10-dit MIC
Table 4. Subgroups of index d of the fundamental group π1(S3 \ 31)
(alias the d-fold coverings over the trefoil knot 3-manifold. The meaning
of symbols is as in Table 2. When the subgroup in question is a subgroup
of the modular group Γ, it is identified as a congruence subgroup or by
its signature NC(g,N, ν2, ν3, [c
Wi
i
]) (see [12] for the meaning of entries).
The permutation group P =SmallGroup(o, k) is abbreviated as (o, k).
As in Table 1, if (i) is true, unless otherwise specified, the graph of
cosets leading to a MIC is the d-simplex [and/or the condition (ii) is
true]. Exceptions (with geometry identified in bold characters) are for
a MIC with (i) true and (ii) false. For index 9 and 10, some subgroups
of large order could not be checked as leading to a MIC or not, they are
not shown in the table. The abbreviation ‘octa’ is for the octahedron,
‘MP’ is for the Mermin pentagram andK33 means three disjoint triangles.