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SL(2, C) Scheme Processing of Singularities in Quantum Computing and
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Citation: Planat, M.; Amaral, M.M.;
Chester, D.; Irwin, K. SL(2,C) scheme
processing of singularities in quantum
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Article
SL(2,C) scheme processing of singularities in quantum
computing and genetics
Michel Planat 1,*,†
, Marcelo M. Amaral 2,†
, David Chester 2,†
and Klee Irwin 2,†
1
Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne-Franche-Comté, F-25044 Besançon, France
2 Quantum Gravity Research, Los Angeles, CA 90290, USA; Marcelo@QuantumGravityResearch.org (M.M.A.);
DavidC@QuantumGravityResearch.org (M.M.A.); Klee@QuantumGravityResearch.org (K.I.)
* Correspondence: michel.planat@femto-st.fr
†
These authors contributed equally to this work.
Abstract: Revealing the time structure of physical or biological objects is usually performed thanks
1
to the tools of signal processing like the fast Fourier transform, Ramanujan sum signal processing
2
and many other techniques. For space-time topological objects in physics and biology, we propose a
3
type of algebraic processing based on schemes in which the discrimination of singularities within
4
objects is based on the space-time-spin group SL(2,C). Such topological objects possess an homotopy
5
structure encoded in their fundamental group and the related SL(2,C) multivariate polynomial
6
character variety contains a plethora of singularities somehow analogous to the frequency spectrum
7
in time structures. Our approach is applied to an Akbulut cork in exotic R4, to an hyperbolic model
8
of topological quantum computing based on algebraic surfaces and to microRNAs in genetics. Such
9
diverse topics reveal the manifold of possibilities of using the concept of a scheme spectrum.
10
Keywords: Finitely generated group; SL(2,C) character variety; algebraic surfaces; schemes; exotic
11
R4; topological quantum computing; microRNAs.
12
0. Introduction
13
In signal processing of a time series, the lines of the Fourier spectrum are described
14
by discontinuities. The approach may be generalized with number theory by taking
15
Ramanujan sums as the building blocks of the signal expansion, e.g. see [1].
16
A more ambitious approach is to use algebraic geometry to reveal the singularities of
17
an object. In recent papers, we explored topics of quantum computing and DNA biology
18
with a common algebraic geometrical tool that we now call SL(2,C) scheme processing.
19
The necessary ingredient is a finitely generated group π expressing the symmetries of
20
the investigated object. The SL(2,C) character variety associated to π is determined and
21
summarized by its Groebner basis G. The multivariate polynomials in G may contain
22
isolated (or non-isolated) singularities that characterize the richness of the object. In this
23
essay, the polynomials are reduced to surfaces living in the 3-dimensional projective space
24
P3(Q) over the rationals. In this way, we may use tools of schemes for the resolution of
25
singularities [2] that are implemented in the software Magma [3].
26
Scheme theory was created by A. Grothendieck to generalize smooth manifolds to
27
algebraic varieties possibly decorated with singularities. For our purpose, it is enough to
28
see a scheme as a geometrical object defined by the vanishing of polynomials defined over
29
an affine (or a projective space) like those living in G.
30
In Section 1, we briefly describe the mathematical formalism used in our paper. It
31
includes the definition of the character variety V representing the finitely generated group
32
π over the group SL(2,C) and how a Groebner basis G is obtained from V in practice.
33
The section mentions the distinction between simple singularities and singularities whose
34
support is not zero dimensional. We then investigate the algebraic geometry of three
35
types of complex objects in physics and biology. The first two objects rely on quantum
36
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computing following our previous papers [4,5]. Section 2 discusses the symmetries and
37
related representations of the Akbulut cork W, a fundamental object in the theory of exotic
38
4-manifolds. Section 3 refers to the symmetries and the related representations of an
39
hyperbolic 3-manifold found in the context of magic states in quantum computing. Section
40
4 investigates a third class of objects, which are a family of biological molecules called
41
microRNAs that regulate the types and amounts of proteins [6]. In Section 5, we provide
42
commentary on our results.
43
1. Theory
44
Details about the theory are described below. The corresponding implementation is
45
available on the Magma software [3].
46
1.1. The SL(2,C) character variety of a finitely generated group and a Groebner basis
47
Let fp be a finitely generated group, we describe the representations of fp in the
48
(double cover alias the group extension of order two of the Lorentz group) SL(2,C), the
49
group of (2× 2) matrices with complex entries and determinant 1. The group SL(2,C) may
50
be seen simultaneously as a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group.
51
Representations of fp in SL(2,C) are homomorphisms ρ : fp → SL(2,C) with char-
52
acter κρ(g) = tr(ρ(g)), g ∈ fp. The set of characters allows to define an algebraic set by
53
taking the quotient of the set of representations ρ by the group SL(2,C), which acts by
54
conjugation on representations [7].
55
Such an algebraic set is called the SL(2,C) character variety of fp. It is made of a
56
sequence of multivariate polynomials called a scheme X. The vanishing of polynomials
57
defines the ideal I(X) of the scheme X. A Groebner basis G(X) is a particular set of the
58
polynomial ring I(X) that has to follow algorithmic rules (similar to the Euclidean division
59
for univariate polynomials).
60
For the effective calculations of the character variety, we make use of a software on
61
Sage [8]. We also need Magma [3] for the calculation of a Groebner basis, at least for 3- and
62
4-letter sequences.
63
1.2. Singularities of an algebraic surface
64
Simple singularities
65
The surfaces S of interest in this case are said to be almost not singular in the sense
66
that they have at worst simple singularities. In Magma, it is referred to a simple or A-D-E
67
singularity if it is an isolated singularity on S which is analytically of the type An, n ≥ 1,Dn,
68
n ≥ 4, E6, E7 or E8.
69
The A-D-E type and the number of simple singularities is reflected in our notation. E.
70
g., S(lA1) means a surface with l singularities of type A1, S(A2) means a surface with a single
71
singularity of type A2 and S(D4) means a surface with a single singularity of type D4 . The
72
Cayley cubic encountered in our previous paper is κ(4A1)
4
(x, y, z) = xyz + x2 + y2 + z2 − 4
73
[5]. The Fricke surface of type D4 is S(D4) = xyz + x2 + y2 + z2 − 8(x + y + z) + 28 [9,
74
Figure 16]. Many other examples can be found in [6].
75
The relevant Magma command for such simply singular surfaces is HasOnlySimpleSin-
76
gularities(S).
77
1.3. Arbitrary singularities
78
Let X ∈ P3(Q) be a singular surface (a surface containing a non zero dimensional
79
singular subset of points). Let Y ∈ P3(Q) be a regular surface (devoid of singularities)
80
above X that is, a representation (in a sense to be qualified) ρ : Y → X. The set of such
81
morphisms ρ belongs to the so-called spectrum SpecX(Y) of Y in X.
82
In our context, the formal desingularization of an (hyper)surface X in P3(Q) is realized
83
with a proper birational map Y → X, with Y is regular. The formal desingularization is
84
realized with the introduction of a formal prime divisor Spec ÔX,p → X, where p ∈ X is a
85
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regular point of codomension 1, OX,p is the structure sheaf at p of X seen as a scheme and
86
the hat means the completion [10].
87
Taking the affine surface S(x, y, z), the related homogeneous polynomial defines a
88
hypersurface X(x, y, z, w) ∈ P3(Q) so that a formal prime divisor is actually a morphism
89
ϕ : Specϕ[[t]] → X defined by the Q-algebra homomorphism ϕ♯ : Q([x, y, z, w]/ < X >→
90
Fϕ[[t]] where Fϕ = Q(s)[α], s a parameter and α is defined by the vanishing of a minimal
91
polynomial.
92
The relevant Magma command for this case is FormallyResolveProjectiveHyperSurface(S:
93
AdjComp := true). Thanks to the setting AdjComp := true, the command returns the number
94
of essential singularities allowing to obtain the formal divisors needed to only compute
95
birational invariants or adjoint spaces [10,11].
Figure 1. The affine singular surface S2(x, y, z) = z4 + 2yz3 + x2 − 6yz− 2x − 8 found in the Groebner
basis for the transcription factor Prdm1 [6, Section 3.1].
96
1.4. Kodaira-Enriques classification
97
Given an ordinary projective surface S in the projective space P3 over a number field,
98
if S is birationally equivalent to a rational surface, the software Magma [3] determines the
99
map to such a rational surface and returns its type within five categories. The returned
100
type of S is P2 for the projective plane, a quadric surface (for a degree 2 surface in P3), a
101
rational ruled surface, a conic bundle or a degree p Del Pezzo surface where 1 ≤ p ≤ 9.
102
A further classification may be obtained for S in P3 if S has at most point singularities
103
(unless the singularities may be formally resolved as we described in the previous sub-
104
section for the case of characteristic zero). Magma computes the type of S (or rather, the
105
type of the non-singular projective surfaces in its birational equivalence class) according
106
to the classification of Kodaira and Enriques [12]. The first returned value is the Kodaira
107
dimension of S, which is −∞, 0, 1 or 2. The second returned value further specifies the type
108
within the Kodaira dimension −∞ or 0 cases (and is irrelevant in the other two cases).
109
Kodaira dimension −∞ corresponds to birationally ruled surfaces. The second return
110
in this case is the irregularity q ≥ 0 of S. So S is birationally equivalent to a ruled surface
111
over a smooth curve of genus q and is a rational surface if and only if q is zero.
112
Kodaira dimension 0 corresponds to surfaces which are birationally equivalent to a K3
113
surface, an Enriques surface, a torus or a bi-elliptic surface.
114
Every surface of Kodaira dimension 1 is an elliptic surface (or a quasi-elliptic surface
115
in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira
116
dimension −∞, 0 or 1.
117
Surfaces of Kodaira dimension 2 are algebraic surfaces of general type.
118
A singular surface
119
Let us illustrate our approach with a selected singular surface encountered in the
120
context of the transcription factor Prdm1 in our recent paper [6, Section 3.1]. The affine
121
surface under question is S2(x, y, z) = z4 + 2yz3 + x2 − 6yz − 2x − 8. It contains 9 essential
122
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singularities in the desingularization. A 3-dimensional plot of S2(x, y, z) is in Figure 1. The
123
surface is a conic bundle of K3 type.
124
The projective surface is S2(x, y, z, w) = z4 + 2yz3 +w2(x2 − 6yz)− 2xw3 − 8w4. There
125
exists 5 distinct types of minimal polynomial. For one of the divisors, the minimal poly-
126
nomial is α2 + 2s + 1 and the prime divisor in the desingularization is the morphism
127
ϕ : Specϕ[[t]] → X defined by
128
x → 1, y → st, z → t, w → αt2 + 0(t4).
The 9 formal morphisms are used to compute the global sections (or adjoints) Homi,j.
129
For i, j < 3, one obtains Hom1,1 = Hom2,1 = Hom2,2 = Hom2,3 = Hom3,1 = Hom3,2 =
130
Hom3,3 = 0, Hom1,2 = [z, w], Hom1,3 = [xz, yz, z2, xw, yw, zw, w2].
131
2. SL(2,C) scheme processing of an Akbulut cork
132
This example belongs to the theory of 4-manifolds and was investigated by the authors
133
in the context of magic states of quantum computing [4]. To our earlier work we add the
134
investigation of SL(2,C) character variety of the relevant fundamental groups. In particular,
135
we put the central object of the exotic R4 manifolds –the Akbulut cork– in a new perspective,
136
by revealing its singularity spectrum.
137
The theory of 4-manifolds is described in books [13–15]. Here we are interested in the
138
decomposition of a 4-manifold into one- and two-dimensional handles as shown in Fig. 2
139
[13, Fig. 1.1 and Fig. 1.2]. Let Bn and Sn be the n-dimensional ball and the n-dimensional
140
sphere, respectively. An observer is placed at the boundary ∂B4 = S3 of the 0-handle B4
141
and watch the attaching regions of the 1- and 2-handles. The attaching region of 1-handle
142
is a pair of balls B3 (the yellow balls), and the attaching region of 2-handles is a framed
143
knot (the red knotted circle) or a knot going over the 1-handle (shown in blue). For closed
144
4-manifolds, there is no need of visualizing a 3-handle since it can be directly attached to
145
the 0-handle. The 1-handle can also be figured out as a dotted circle S1 × B3 obtained by
146
squeezing together the two three-dimensional balls B3 so that they become flat and close
147
together [14, p. 169] as shown in Fig. 2b.
148
For the attaching region of a 2- and a 3-handle one needs to enrich our knowledge by
149
introducing the concept of an Akbulut cork to be described in the next paragraph [4, Figure
150
3].
151
Figure 2. (a) Handlebody of a 4-manifold with the structure of 1- and 2-handles over the 0-handle B4,
(b) the structure of a 1-handle as a dotted circle S1 × B3, (c) an Akbulut cork W = 946(−1, 1).
2.1. Akbulut cork
152
A Mazur manifold is a contractible, compact, smooth 4-manifold (with boundary) not
153
diffeomorphic to the standard 4-ball B4 [13]. Its boundary is a homology 3-sphere. If we
154
restrict to Mazur manifolds that have a handle decomposition into a single 0-handle, a
155
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single 1-handle and a single 2-handle then the manifold has to be of the form of the dotted
156
circle S1 × B3 (as in Fig. 2b) (right) union a 2-handle. The simplest object of this type is the
157
Akbulut cork shown in Fig. 2c [16,17].
158
Given p, q, r (with p ≤ q ≤ r), the Brieskorn 3-manifold Σ(p, q, r) is the intersection
159
in the complex 3-space C3 of the 5-dimensional sphere S5 with the surface of equation
160
zp1 + z
q
2 + z
r
3 = 0. The smallest known Mazur manifold is the Akbulut cork W and its
161
boundary is the Brieskorn homology sphere Σ(2, 5, 7). The Akbulut cork has a simple
162
definition in terms of the framings ±1 of (−3, 3,−3) pretzel knot also called K = 946 [18,
163
Fig. 3]. It has been shown that ∂W = Σ(2, 5, 7) = K(1, 1) and W = K(−1, 1).
164
An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic
165
to the Euclidean space R4. An exotic R4 is called small if it can be smoothly embedded as
166
an open subset of the standard R4 and is called large otherwise. Here we are concerned
167
with an example of a small exotic R4.
168
According to [18], there exists an involution f : ∂W → ∂W that surgers the dotted
169
1-handle S1 × B3 to the 0-framed 2-handle S2 × B2 and back, in the interior of W. There
170
exists a smooth contractible 4-manifold V with ∂V = ∂W, such that V is homeomorphic
171
but not diffeomorphic to W relative to the boundary [16, Theorem 1]. This leads us to our
172
next paragraph.
173
2.2. The manifold W̄ mediating the Akbulut cobordism between exotic manifolds V and W
174
A cobordism between two oriented m-manifolds M and N is any oriented (m + 1)-
175
manifold W0 such that the boundary is ∂W0 = M̄ ∪ N, where M appears with the reverse
176
orientation. The cobordism M × [0, 1] is called the trivial cobordism. Next, a cobordism W0
177
between M and N is called an h-cobordism if W0 is homotopically like the trivial cobordism.
178
The h-cobordism due to S. Smale in 1960, states that if Mm and Nm are compact simply-
179
connected oriented M-manifolds that are h-cobordant through the simply-connected (m +
180
1)-manifold Wm+1
0
, then M and N are diffeomorphic [15, p. 29].
181
However this theorem fails in dimension 4. If M and N are cobordant 4-manifolds,
182
then N can be obtained from M by cutting out a compact contractible submanifold W and
183
gluing it back in by using an involution of ∂W.
184
The h-cobordism under question in our example may be described by attaching an
185
algebraic cancelling pair of 2- and 3-handles to the interior of Akbulut cork W. The 4-
186
manifold W̄ mediating V and W resembles the Akbulut cork with the dot replaced by a
187
0-surgery The manifold under question is nothing but L7a6(0, 1)(0, 1)] (see [16, p. 355] or
188
[4, Figure 3c]).
189
2.3. The character variety for an Akbulut cork W
190
With Snappy, we find that the fundamental group ruling the Akbulut cork W =
191
946(−1, 1) is the two-generator group
192
π1(W) =
〈
a, b|aBAb2 ABabaBABab, a3BAb3 AB
〉
, A = a−1, B = b−1.
(1)
With Sage software in Reference [8], we compute the corresponding character variety.
193
Then, from Magma [3], the Groebner basis is found in the form
194
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Figure 3. The surface SW(x, y, z) found within the Groebner basis of the SL(2,C) character variety
for the Akbulut cork W.
GW(x, y, z) = z(z − 2)[x + y + f1(z)][y2 − y + f2(z][y + f3(z)] f4(z),
f1(z) = −171/34 z8 + 913/34 z7 − 2429/34 z6 + 4733/34 z5 − 2443/17 z4
−224/17 z3 + 1847/17 z2 − 1103/34 z − 1,
f2(z) = −95/34 z8 + 477/34 z7 − 1221/34 z6 + 2331/34 z5 − 1089/17 ∗ z4
−285/17 z3 + 858/17 z2 − 337/34 z − 1
f3(z) = −71/17 z7 + 376/17 z6 − 997/17 z5 + 1942/17 z4 − 1993/17 z3
−188/17 z2 + 1477/17 z − 436/17,
f4(z) = z7 − 5 z6 + 13 z5 − 25 z4 + 24 z3 + 4 z2 − 18 z + 5.
(2)
The factor containing f1(z) in Equation 3 is the singular surface SW(x, y, z) shown in
195
Figure 3. It is a rational scroll and a surface of a general type. The factor containing f2(z)
196
is an hyperelliptic function of genus 7, discriminant ≈ 1.327502101 and points at infinity
197
(1, 0, 0) and (1,−95/34, 0). The factor containing f3(z) is an ordinary curve. The factor
198
containing f4(z) is a seventh-order polynomial.
199
Formal desingularization of the surface SW(x, y, z)
200
The formal desingularization of a hypersurface X in the three-dimensional projective
201
space P3Q over the rationals Q is described in [10] and can be explicitely given with Magma
202
[3, Section 122.5.3].
203
To the surface X = SW(x, y, z) we associate the degree 8 homogeneous polyno-
204
mial SW(x, y, z, w) = x + y + f1(z, w) ∈ Q(x, y, z, w) , with f1(z, w) = −171/34z8 +
205
913/34 wz7 + · · · − 1103/343 w7z − w8.
206
For the projective surface X = SW(x, y, z, w), the two essential (over the three) singular
207
morphisms are
208
x → 1
y → st − 1
z → t
w → αt + O(t8),
(3)
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209
x → 1
y → s
z → 34/171 (s + 1)t7 + O(t8)
w → O(t8),
(4)
where α has minimal polynomial −s − α8 f1(α−1).
210
These formal morphisms are used to compute the global sections (or adjoints) Homi,j.
211
For the surface SW(x, y, z) one gets
212
Hom1,1 = Hom2,1 = Hom2,2 = 0, Hom1,2 = [z6, z5w, z4w2, z3w3, z2w4, zw5, w6] · · ·
213
2.4. The character variety for the mediating manifold W̄
214
The fundamental group π1(W̄) of the h-cobordism W̄ is as follows [4]
215
π1(W̄) =
〈
a, b|a3b2 AB3 Ab2, (ab)2aB2 Ab2 AB2
〉
.
(5)
The cardinality structure of subgroups of this fundamental group is
216
ηd[π1(W̄)] = ηd[π1(W)] = [1, 0, 0, 0, 1, 1, 2, 0, 0, 1, 0, 5, 4, 9, 7, 1 · · · ],
where the bold digits refers to our investigation in [4, Table 1] about the existence
217
of a finite geometry obtained with a subgroup of the corresponding index. The smallest
218
case occurs at index 10 and the geometry is the Mermin pentagram, a type of contextual
219
geometry.
220
As before, with Sage software in Reference [8], we compute the corresponding charac-
221
ter variety. Then, from Magma, the Groebner basis is found in the form
222
GW̄ = (z − 2)(z2 − z − 1)[x + g1(z)][y + g2(z)]g3(z),
g1(z) = −z9 + 3z8 + 5z7 − 18z6 − 10z5 + 43z4 − 2z3 − 27z2 + 3z + 4,
g2(z) = −z8 + 2z7 + 6z6 − 11z5 − 15z4 + 24z3 + 9z2 − 10z − 2,
g3(z) = z7 + z6 − 5z5 − 6z4 + 6z3 + 5z2 − 2z − 1.
(6)
Although the card seq for manifolds W and W̄ are the same, we clearly see that
223
the character variety are distinct. In the later case, it contains two ordinary curves and
224
polynomials of degree 1, 2 and 7.
225
3. SL(2,C) scheme processing in topological quantum computing
226
In this section, we are interested in the SL(2,C) character variety of the fundamental
227
group of (hyperbolic) 3-manifold L10n46 (alias otet0800002). This manifold describes the
228
4-fold (irregular) covering of the figure-eight knot 41 = K4a1 [19, Table 2]. It is connected
229
to the magic state describing the contextual geometry of two-qubits (e.g. [19, Figure 1] or
230
[20, Table 1]). The basic idea has been to put magic states and informationally complete
231
POVMs under the same hat [21]. Thanks to the character variety and the related algebraic
232
surfaces , we can add the topological aspect to the previous description.
233
The Groebner basis of the character variety for the fundamental group of 3-manifold
234
L10n46 may be obtained with Magma. The 3-generator fundamental group is
235
π1(L10n46) = ⟨a, b, c|abAcBac, abbCBccBBBACbcbC⟩.
(7)
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The character variety GL10n46(e, f , g, h)(x, y, z) is now seven-dimensional as in [6]
236
and selected choices of parameters e, f , g and h allow us to determine the algebraic surfaces
237
within the Groebner basis.
238
For instance, we find
239
GL10n46(0, 0, 0, 0) = yS1(x, y, z)S2(x, y, z)S3(x, y, z) · · ·
S1(x, y, z) = f
(A1)
2,{} (x, y, z) = xyz + xz
2 + x2 + y2 + z2 + yz − x − 6,
S2(x, y, z) = xz2 + yz − x − 2,
S3(x, y, z) = −x2z2 + 2x2 + y2 + z2 + 2x − 4.
(8)
Figure 4. Left: the (degree 3) del Pezzo surface S1(x, y, z) = f
(A1)
2,{} (x, y, z) = xyz + xz
2 + x2 + y2 +
z2 + yz − x − 6 . Middle: the (rational scroll) surface S2(x, y, z) = xz2 + yz − x − 2 . Right: the (del
Pezzo degree 4) surface S3(x, y, z) = −x2z2 + 2x2 + y2 + z2 + 2x − 4.
The first surface S1(x, y, z) = f
(A1)
2,{} in the product (8) has a single simple singularity of
240
type A1 whose reduced singular subscheme is of degree 2 with a vanishing support.
241
The second surface S2(x, y, z) is a singular rational scroll with a single essential singu-
242
larity. The corresponding singular morphism Spec Fϕ[[t]] → X = S2(x, y, z, w) is
243
x → 1
y → s
z → t
w → ((−2s3 − 6s)/(s2 + 4)α+ (−2s2 − 4)/(s2 + 4))αt + O(t3),
(9)
where α has minimal polynomial α2 − sα− 1.
244
The third surface S3(x, y, z) = −x2z2 + 2x2 + 2x + y2 + z2 − 4 is a singular surface of
245
the degree 4 del Pezzo type. There are 4 essential singularities. We retain one of the two
246
singularities with a nontrivial minimal polynomial. The corresponding singular morphism
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Spec Fϕ[[t]] → X = S1(x, y, z, w) is
248
x → 1
y → s
z → t
w → αt − 1/(s4 + 4s2 + 4)t2 + (−1/2 s4 + 9/2)/(s6 + 6s4 + 12s2 + 8)αt3) + O(t5)
(10)
where α has minimal polynomial α2 − 1/(s2 + 2).
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All three affine surfaces are shown in Figure 4.
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Version January 20, 2023 submitted to Axioms
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4. SL(2,C) scheme processing in microRNAs
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In this section, we focus on the SL(2,C) character varieties attached to microRNAs
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(miRNAs for short). This case was already tackled in our recent paper [6].
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The miRNAs are short (approximately 22 nt long) single-stranded RNA molecules
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playing a fundamental role in the expression and regulation of genes by targeting specific
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messenger RNAs (mRNAs) for degradation or translational repression. The genes encoding
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miRNAs are much longer than the processed mature miRNA molecule. There are pre-
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miRNAs comprising of approximately 70-nucleotides in length which are folded into
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imperfect stem-loop structures.
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Each miRNA is synthesized as an miRNA duplex comprised of two strands (-5p and -
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3p). However, only one of the two strands becomes active, which is selectively incorporated
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into the RNA-induced silencing complex in a process known as miRNA strand selection
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[22,23]. For details about the miRNA sequences, we use the Mir database [24,25].
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Disregulation of miRNAs may lead to a disease like cancer. A key microRNA known
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as an oncomir (involved in immunity and cancer) is mir-155 [6].
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Here, we select examples of human miRNAs from the perspective of evolution. The
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generator of the group π to be considered is a short (about 8-letter long) seed made of two to
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four distinct bases in the set {A,U,G,C}. Most of the time, π is close to a free group of rank
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equal the number of distinct bases in the seed minus one. This point can be checked by the
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cardinality structure of conjugacy classes subgroups of π (denoted card seq). Exceptions to
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this rule and the occurrence of singularities (isolated or not) in the corresponding character
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variety built from π, is a witness of a potential disease. Unlike the case of transcription
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factors, until now, singularities possibly found with microRNAs are isolated singularities.
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According to Reference [26, Table 3], the slowest evolving miRNA gene is hsa-mir-503
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(the notation hsa is for the human specie). It is known that mir-503 regulates gene expression
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from different aspects of pathological processes of diseases, including carcinogenesis,
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angiogenesis, tissue fibrosis and oxidative stress [27]. The seed region for mir-503-5p is
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AGCAGCGG and the corresponding Groebner basis for parameters (e, f , g, h) = (0, 0, 0, 0)
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is very simple: Gmir−503−5p(0, 0, 0, 0) = κ
(4A1)
4
(x, y, z), as shown in Figure 5 (Left).
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For (e, f , g, h) = (1, 1, 0, 0), Gmir−503−5p(1, 1, 0, 0) = −3xyzκ3(x, y, z), with κ3(x, y, z) is
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the Fricke surface found in [5, Section 3.3]. For (e, f , g, h) = (1, 1, 1, 1), there are many more
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polynomials. One of them defines the Fricke surface xyz + x2 + y2 + z2 − 2x − y − 2. The
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considered seed region for mir-503-3p is GGGUAUU. The surfaces in the Groebner basis
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are very simple in this case and not even simple singularities lie in them.
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One of the fastest evolving microRNA is mir-214 [26, Table 3]. First mir-214 was
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reported to promote apoptosis in HeLa cells. Presently, mi-214 is implicated in an extensive
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range of conditions such as cardiovascular diseases, cancers, bone formation and cell
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differentiation [28]. For mir-214-5p and the seed sequence GCCUGU, one finds the surface
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f (A2)
1,{1:0:0:0}(x, y, z) = xyz + y
2 + z2 − 4 within Gmir−214−5p(0, 0, 0, 0), as shown in Figure
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5. A surface of the same type f (A2)
1,{1:0:0:0}(x, y, z) = xyz + y
2 + z2y − 2z − 1 is found in
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Gmir−214−5p(1, 1, 1, 1). For a longer seed, surfaces are not found to contain singularities.
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5. Conclusion
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Over the last few years, the authors of this article have found that some mathematical
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techniques employed for quantum information processing and quantum computing may
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also apply to biology at the genome scale. More precisely, group theory and representations
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of symmetries with characters of finite groups have been used for topological quantum
296
computing (TQC) [19] or elementary particles [29], and the the encoding of proteins [30].
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Methods for dealing with infinite groups and SL(2,C) representations of such groups in
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TQC papers [5,31] were similarly employed for transcription factors [6] and miRNAs [6].
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Our efforts in this paper are belonging to the field of scheme processing, where
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the desingularization of discontinuities of algebraic surfaces is a spectrum. The scheme
301
spectrum is a well known concept in commutative algebra. The prime spectrum of a ring
302
Version January 20, 2023 submitted to Axioms
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Figure 5. Left: the Cayley cubic κ4(x, y, z) found in the character variety for the slowest evolving
miRNA gene hsa-mir-503. The surface f (A2)
1,{1:0:0:0}(x, y, z) found in the character variety Gmir−214−5p of
the fast evolving gene hsa-mir-214.
is the set of prime ideals of the ring R and denoted by Spec(R). The sheaf of rings O
303
is the relevant algebraic geometrical notion introduced by Grothendieck to develop this
304
field [2,32]. We touched this important concept of schemes while investigating non-zero
305
dimensional sets of singularities in surfaces belonging to the SL(2,C) character variety of
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an infinite group. We are fortunate that the software Magma is designed to implement
307
schemes in a variety of applications (curves, surfaces and more). Another computer algebra
308
system with similar facilities is Singular [33].
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We have no doubt that scheme theory will play and increasing role in our future efforts
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of relating quantum concepts and biology. In the future, this may be applicable to the field
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of ‘quantum consciousness’ [34].
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Author Contributions: “Conceptualization, M.P.; methodology, M.P and M.A.; software, M.P.;
313
validation, M.A., D.C. and K.I.; formal analysis, D.C., M.P. ; investigation, D.C., M.A., M.P. ; resources,
314
K.I.; data curation, M.P.; writing—original draft preparation, M.P.; writing—review and editing,
315
M.P. and M.A.; visualization, M.A.; supervision, M.P.; project administration, M.P., K.I.; funding
316
acquisition, K.I. All authors have read and agreed to the published version of the manuscript”.
317
Funding: “This research received no external funding”
318
Informed Consent Statement: “Not applicable”
319
Data Availability Statement: “Data are available from the authors after a reasonable demand”
320
Conflicts of Interest: “The authors declare no conflict of interest”
321
MDPI Multidisciplinary Digital Publishing Institute
322
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