We recently proposed that topological quantum computing might be based on SL(2,C) representations of the fundamental group π1(S3\K) for the complement of a link K in the three-sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke surface κd=xyz−x2−y2−z2+d is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead link 521, the Berge link 622 or the double-eight link 623) or the link 723, the V for those links contains the reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd (d<4), with |x|,|y|,|z|≤2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

### About Klee Irwin

**Klee Irwin is an author, researcher and entrepreneur who now dedicates the majority of his time to Quantum Gravity Research (QGR), a non-profit research institute that he founded in 2009. The mission of the organization is to discover the geometric first-principles unification of space, time, matter, energy, information, and consciousness. **

**As the Director of QGR, Klee manages a dedicated team of mathematicians and physicists in developing emergence theory to replace the current disparate and conflicting physics theories. Since 2009, the team has published numerous papers and journal articles analyzing the fundamentals of physics. **

**Klee is also the founder and owner of Irwin Naturals, an award-winning global natural supplement company providing alternative health and healing products sold in thousands of retailers across the globe including Whole Foods, Vitamin Shoppe, Costco, RiteAid, WalMart, CVS, GNC and many others. Irwin Naturals is a long time supporter of Vitamin Angels, which aims to provide lifesaving vitamins to mothers and children at risk of malnutrition thereby reducing preventable illness, blindness, and death and creating healthier communities.**

**Outside of his work in physics, Klee is active in supporting students, scientists, educators, and founders in their aim toward discovering solutions to activate positive change in the world. He has supported and invested in a wide range of people, causes and companies including Change.org, Upworthy, Donors Choose, Moon Express, Mayasil, the X PRIZE Foundation, and Singularity University where he is an Associate Founder.**

Amaral, M.M.; Irwin, K. Fricke

Topological Qubits. Quantum Rep.

2022, 4, 523–532. https://doi.org/

10.3390/quantum4040037

Academic Editor: Antonio Manzalini

Received: 7 October 2022

Accepted: 9 November 2022

Published: 14 November 2022

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quantum reports

Article

Fricke Topological Qubits

Michel Planat 1,*,†

, David Chester 2,†

, Marcelo M. Amaral 2,†

and Klee Irwin 2,†

1

Institut FEMTO-ST CNRS UMR 6174, Université de Bourgogne/Franche-Comté, 15 B Avenue des

Montboucons, F-25044 Besançon, France

2 Quantum Gravity Research, Los Angeles, CA 90290, USA

* Correspondence: michel.planat@femto-st.fr

†

These authors contributed equally to this work.

Abstract: We recently proposed that topological quantum computing might be based on SL(2,C)

representations of the fundamental group π1(S3 \ K) for the complement of a link K in the three-

sphere. The restriction to links whose associated SL(2,C) character variety V contains a Fricke

surface κd = xyz− x2 − y2 − z2 + d is desirable due to the connection of Fricke spaces to elementary

topology. Taking K as the Hopf link L2a1, one of the three arithmetic two-bridge links (the Whitehead

link 521, the Berge link 6

2

2 or the double-eight link 6

2

3) or the link 7

2

3, the V for those links contains the

reducible component κ4, the so-called Cayley cubic. In addition, the V for the latter two links contains

the irreducible component κ3, or κ2, respectively. Taking ρ to be a representation with character κd

(d < 4), with |x|, |y|, |z| ≤ 2, then ρ(π1) fixes a unique point in the hyperbolic space H3 and is a

conjugate to a SU(2) representation (a qubit). Even though details on the physical implementation

remain open, more generally, we show that topological quantum computing may be developed from

the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI

equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example.

Keywords: topological quantum computing; SL(2,C) character variety; knot theory

1. Introduction

Building a quantum computer is still challenging. However, progress has been made

using natural and artificial atoms [1], superconducting technology [2] and other physical

techniques [3,4]. One of the greatest challenges involved with constructing quantum

computers is controlling or removing quantum decoherence. One possible solution is to

create a topological quantum computer.

The paper describes progress towards an understanding and possibly an implemen-

tation of quantum computation based on algebraic surfaces. In the orthodox acceptation,

a topological quantum computer deploys two-dimensional quasiparticles called anyons

that are braids in three dimensions. The braids lead to logic gates used for computation.

The topological nature of the braids makes the quantum computation less sensitive to the

decoherence errors than in a standard quantum computer [5,6]. One theoretical proposal of

universal quantum computation is based on Fibonacci anyons that are non-Abelian anyons

with fusion rules. In particular, a fractional quantum Hall device would, in principle, realize

a topological qubit. Owing to the lack of evidence that such quantum Hall-based anyons

have been obtained, other theoretical proposals are worthwhile to develop. A recent paper

of our group proposed a correspondence between the fusion Hilbert space of Fibonacci

anyons and the tiling two-dimensional space of the one-dimensional Fibonacci chain [7].

In this paper, following our recent proposal [8] (see also [9]), we propose a non-

anyonic theory of a topological quantum computer based on surfaces in a three-dimensional

topological space. Such surfaces are part of the SL(2,C) character variety underlying the

symmetries of a properly chosen manifold. In our earlier work, we were interested in

basing topological quantum computing on three- or four-manifolds defined from the

Quantum Rep. 2022, 4, 523–532. https://doi.org/10.3390/quantum4040037

https://www.mdpi.com/journal/quantumrep

Quantum Rep. 2022, 4

524

complement of a knot or link. In [10,11], our goal was to define informationally complete

quantum measurements from three-manifolds and, in [12], from four-manifolds, seeing the

embedding four-dimensional ‘exotic’ space R4 of the manifold as a the physical Euclidean

space–time. In the later paper, exotism means that one can define homeomorphic but non-

diffeomorphic four-dimensional manifolds to interpret a type of ‘many-world’ quantum

measurements.

Our concepts in [8] and in the present paper are different in the sense that the SL(2,C)

character variety is the three-dimensional locus of the supposed qubit prior to its measure-

ment. The Lorentz group SL(2,C) reads the symmetries of the selected topology like that

of the punctured torus, the quadruply punctured sphere or the topology obtained from the

complement of a knot or a link. Our work in [8] focused on the complement of the Hopf

link—the linking of two unknotted curves—where the character variety consists of the Cay-

ley cubic κ4(x, y, z). Here, we took the broader context of Fricke surfaces, whose compact

bounded component consists of the SU(2) representations [13]. Such representations are

our proposed model of the topological qubits.

In Section 2, we recall the definition of the SL(2,C)-character variety for a manifold

M whose fundamental group is π1(M) and the method used to build it in an explicit way.

In Section 3, we focus on the character variety κ2(x, y, z) for the fundamental group

F2 (the free group of rank 2) of the once-punctured torus S1,1 and on the character variety

κ4(x, y, z) attached to the fundamental group of the Hopf link L2a1. The former case is

found to be related to the two-bridge link L7a4. The role of the extended mapping class

group Mod±(S1,1) on a character variety of type κd(x, y, z), d ∈ C, is emphasized. We also

introduce the concept of a topological qubit associated to the bounded SU(2) component

of the surface κ2(x, y, z).

In Section 4, we focus on the character variety Va,b,c,d(x, y, z) for the fundamental group

(the free group of index 3) F3 of the quadruply punctured sphere S4,2. In particular, we

recall the conditions that separate the compact (and bounded) SU(2) component and the

non-compact SL(2,R) component. The investigation of the coverings of the four-manifold

Ẽ8 (the Kodaira singular fiber I I∗) allows for an application of the theory. In the same

section, we describe the Riemann–Hilbert correspondence for the case of S4,2, as well as the

so-called Painlevé-Okomoto correspondence. The Painlevé VI equation plays a special role.

In Section 5, we apply some perspectives of the present research toward topological

quantum computing related to cosmology.

2. The SL(2,C)-Character Variety of a Manifold M

Let π1 be the fundamental group of a topological surface S. We describe the represen-

tations of π1 in the Lorentz group SL(2,C), the group of (2× 2) matrices with complex

entries and determinant 1. Such a group contains representations as degrees of freedom for

all quantum fields and is the gauge group for the Einstein–Cartan theory, which contains

the Einstein–Hilbert action and Einstein’s field equations [14]. Topological formulations of

gravity have been introduced [15–17] and the relationship of entanglement with spacetime

has been articulated [18,19], which motivates the exploration of SL(2,C) character varieties

for quantum computation.

Representations of π1 in SL(2,C) are homomorphisms ρ : π1 → SL(2,C) with char-

acter κρ(g) = tr(ρ(g)), g ∈ π1. The set of characters allows us to define an algebraic set

by taking the quotient of the set of representations ρ by the group SL2(C), which acts by

conjugation on representations [13,20].

Below, we need the distinction between the two real forms SU(2) and SL(2,R) of

the group SL(2,C). The SU(2) representations are those that fix a point in the three-

dimensional hyperbolic space H3 and SL(2,R) representations are those that preserve a

two-dimensional hyperbolic space H2 in H3, as well as an orientation of H2. Both real forms

of SL(2,C) play an important role in our attempt to define and stabilize the potential Fricke

topological qubits.

Quantum Rep. 2022, 4

525

A Sage Program for Computing the SL(2,C)-Character Variety of π1(S)

The SL(2,C) character variety of a manifold M defined in SnapPy may be calculated

in Sage using a program [21] written by the last author of [20] as follows:

from snappy import Manifold

M = Manifold(‘M’)

G = M.fundamental_group()

I = G.character_variety_vars_and_polys(as_ideal=True)

I.groebner_basis()

In some cases, a Groebner basis is not obtained from Sage. One may also use

Magma [22] to obtain the Groebner basis or a small basis with the shortest length of

the ideal I.

3. The Cubic Surface κd(x, y, z) and Two-Bridge Links

Following [23], in this section, we describe the special case of representations for the

punctured torus S1,1 and the relevance of the extended mapping class group Mod±(S1,1)

in its action on surfaces of type κd(x, y, z), d ∈ C. Then, we find that surfaces κ2(x, y, z) and

κ3(x, y, z) are contained in the character variety for the fundamental group of links L7a4

and L6a1, respectively. The SU(2) representations and the concept of a Fricke topological

qubit is outlined.

3.1. The SL(2,C)-Character Variety for a Once Punctured Torus

Let us take the example of the punctured torus T1,1 whose fundamental group π1

is the free group F2 = 〈a, b|∅〉 on two generators a and b. The boundary component of

T1,1 is a single loop around the puncture expressed by the commutator [a, b] = abAB with

A = a−1 and B = b−1. We introduce the traces

x = tr(ρ(a)), y = tr(ρ(b)), z = tr(ρ(ab)).

The trace of the commutator is the surface [13,23]

tr([a, b]) = κ2(x, y, z) = x2 + y2 + z2 − xyz− 2.

Another noticeable surface is obtained from the character variety attached to the

fundamental group of the Hopf link L2a1 that links two unknotted curves. For the Hopf

link, the fundamental group is

π1(S3 \ L2a1) = 〈a, b|[a, b]〉 = Z2,

and the corresponding character variety is the Cayley cubic [8]

κ4(x, y, z) = x2 + y2 + z2 − xyz− 4.

Both surfaces κ4 and κ2 are shown in Figure 1.

Surfaces κ2 and κ4 have been obtained from two different mathematical concepts, from

topological and algebraic concepts in dimension two, respectively. To relate them, one

makes use of the Dehn–Nielsen–Baer theorem applied to the once-punctured torus [24].

According to this theorem, for a surface of genus g ≥ 1, we have

Mod±(Sg) ∼= Out(π1(Sg)),

where the mapping class group Mod(S) denotes the group of isotopy classes of orientation-

preserving diffeomorphisms of S (that restrict to the identity on the boundary ∂S if ∂S

6= ∅),

the extended mapping class group Mod±(S) denotes the group of isotopy classes of all

homeomorphisms of S (including the orientation-reversing ones) and Out(π1) denotes the

outer automorphism group of π1(S). This leads to the (topological) action of Mod± on the

punctured torus as follows:

Quantum Rep. 2022, 4

526

Mod±(S1,1) = Out(F2) = GL(2,Z).

(1)

Figure 1. Top left: the Cayley cubic κ4(x, y, z), Top right: the surface κ2(x, y, z), Down, the surface

κd(x, y, z) with d = − 116 .

The automorphism group Aut(F2) acts by composition on the representations ρ and

induces an action of the extended mapping class group Mod± on the character variety by

polynomial diffeomorphisms of the surface κd defined by [25]

κd(x, y, z) = xyz− x2 − y2 − z2 + d.

(2)

3.2. The Surface κ2, the Link L7a4 = 723 and Fricke Topological Qubits

The surface κ2 corresponds to representations ρ : π1(κ2)→ SL(2,C) of the group ([25]

Section 4.2)

π1(κ2) =

〈

a, b|[a, b]4

〉

.

(3)

Since the surface κ4(x, y, z) is the character variety of the Hopf link, we would also like

to obtain a link whose character variety contains the surface κ2(x, y, z). Making use of the

Thistlethwaite link table [26], we find that the only two-bridge link that has this property is

the link L7a4 = 723; see Figure 2. Taking 0-surgery on both cusps of L7a4, Snappy calculates

the fundamental group as

π1(S3 \ L7a4(0, 1)(0, 1)) =

〈a, b|aBAbabAbabABaBAB, abAbaabAbabABBBBBBAb〉.

The corresponding Groebner base for the character variety is

κL7a4(x, y, z) = xyz(z2 − 2)κ4(x, y, z)κ2(x, y, z),

whose factorization contains both surfaces κ4(x, y, z) and κ2(x, y, z).

Quantum Rep. 2022, 4

527

Figure 2. The 0-surgery on both pieces of links L7a4 in (a) and L6a1 in (b). The Groebner base for the

corresponding character varieties contains the surfaces κ2(x, y, z) and κ4(x, y, z) for the former case,

and κ3(x, y, z) for the latter case.

Topological Qubits from κ2(x, y, z)

Qubits are the elements of group SU(2). There is an interesting connection of the

group π1(κ2) in Equation (3) to SU(2) representations.

According to [25] [Theorem 1.1], there exists a representation ρ : π1(κ2) → SL(2,C)

such that the closure of the orbit of its conjugacy class κ(ρ) under the action of the extended

mapping class group Out(F2) in Equation (1) contains the whole set of SU(2) representa-

tions of π1(κ2). The subset of the real surface κ2(x, y, z) consisting of SU(2) representations

is the unique bounded connected component of κ2(x, y, z) homeomorphic to a sphere; see

Figure 1 (Right).

The bounded component is invariant under the mapping class Ψ =

(

2 1

1 1

)

and there

are two fixed points of the polynomial transformation fΨ(x, y, z) = (z, yz− x, z(yz− x)− y)

made of points (x, x/(x− 1), x) with irrational values x ∼ 0.52 and x ∼ −1.1 ([25], p. 19).

3.3. The Surface κ3(x; y, z) and the Link 623 = L6a1

We would like to obtain a link whose character variety contains the surface κ3(x, y, z).

Making use of the Thistlethwaite link table [26], we find that the only two-bridge link that

has this property is the link 623 = L6a1; see Figure 2. Taking 0-surgery on both cusps of

L6a1, Snappy calculates the fundamental group as

π1(S3 \ L6a1(0, 1)(0, 1)) =

〈a, b|abbaBAbaabAB, aBabABabaBAb〉.

The corresponding Groebner base for the character variety is

κL6a1(x, y, z) = xκ3(x, y, z)(x2 + y2 − xyz)(−xy2z+ y3 + x2y+ xz− 2y) ∗ f9(x, y, z)

whose factorization contains surface κ3(x, y, z) and a ninth-order trivariate polynomial

f9(x, y, z) not made explicit here.

4. The Fricke Cubic Surface and Three-Bridge Links

Our main object in this section is the four-punctured sphere S4,2, for which, the

fundamental group is the free group F3 of rank three whose character variety generalizes

the Fricke cubic surface (2) to the hypersurface Va,b,c,d(C) in C7. It is shown how this

hypersurface is realized in the variety of a covering of index 6 of the four-manifold Ẽ8, the

0-surgery on all circles of the Borromean rings BR0. The Okamoto–Painlevé correspondence

is re-examined in terms of Dynkin diagrams of the appropriate four-manifolds.

Quantum Rep. 2022, 4

528

4.1. The SL(2,C)-Character Variety for the Quadruply Punctured Sphere S4,2

We follow the work of references [13,25,27].

The fundamental group for S4,2 can be expressed in terms of the boundary components

A, B, C and D as π1(S4,2) = 〈A, B, C, D|ABCD〉 ∼= F3.

A representation π1 → SL(2,C) is a quadruple

α = ρ(A), β = ρ(B) γ = ρ(C), δ = ρ(D) ∈ SL(2,C) where αβγδ = I.

Let us associate the seven traces

a = tr(ρ(α)), b = tr(ρ(β)), c = tr(ρ(γ)), d = tr(ρ(δ))

x = tr(ρ(αβ)), y = tr(ρ(βγ)), z = tr(ρ(γα)),

where a, b, c and d are boundary traces and x, y and z are traces of elements AB, BC and

CA representing simple loops on S4,2.

The character variety for S4,2 satisfies the equation ([13] Section 5.2, [25] Section 2.1, [27]

Section 3B, [28] Equation (1.9) or [29] Equation (39))

Va,b,c,d(C) = Va,b,c,d(x, y, z) = x2 + y2 + z2 + xyz− θ1x− θ2y− θ3z− θ4 = 0

(4)

with θ1 = ab+ cd, θ2 = ad+ bc, θ3 = ac+ bd and θ4 = 4− a2 − b2 − c2 − d2 − abcd.

4.2. A Compact Component of SL(2,R)

As shown in the previous section, for the real surface κ2(x, y, z), the compact compo-

nent is made of SU(2) representations.

For the real surface Va,b,c,d(R), there exists a compact component if and only if [27],

Proposition 1.4

∆(a, b, c, d) =

(2(a2 + b2 + c2 + d2)− abcd− 16)2 − (4− a2)(4− b2)(4− c2)(4− d2) > 0

and 16− abcd− 2(a2 + b2 + c2 + d2) > 0

(5)

When Equation (5) is satisfied and (a, b, c, d) ∈ (−2, 2), then Va,b,c,d(R) contains a

compact component made of SL(2,R) representations. Otherwise, each element of the

component is the character of an SU(2) representation; see also ([23], Theorem 9.6).

The former case occurs in the following example, a, b, c = 32 and d = −

3

2 , for which,

the surface is Va,b,c,d = x2 + y2 + z2 + xyz− 116 ([27], p. 102). A sketch of this surface is

given in Figure 1.

4.3. The SL(2,C) Character Variety for the Manifold Ẽ8 and for Its Covering Manifolds

In a recent paper ([8], Section 3.2), we noticed connections between the coverings of

the manifold Ẽ8 and the matter of topological quantum computing. The affine Coxeter–

Dynkin diagram Ẽ8 corresponds to the fiber I I∗ in Kodaira’s classification of minimal

elliptic surfaces ([30], p. 320). Alternatively, one can see Ẽ8 as the 0-surgery on the trefoil

knot 31. The boundary of the manifold associated to Ẽ8 is the Seifert fibered toroidal

manifold [31,32].

The coverings of the fundamental group π1(S4 \ 31(0, 1)) are fundamental groups of

the manifolds in the following sequence:

[Ẽ8, Ẽ6, {D̃4, Ẽ8}, {Ẽ6, Ẽ8}, Ẽ8, {BR0, D̃4, Ẽ6}, {Ẽ8}, {Ẽ6}, {D̃4, Ẽ8}, Ẽ6, · · · ]

The subgroups/coverings are fundamental groups for Ẽ8 Ẽ6, D̃4 or BR0, where BR0 is

the manifold obtained by 0-surgery on all circles of Borromean rings.

Quantum Rep. 2022, 4

529

A Groebner base for the SL(2,C) character variety of π1(Ẽ8) is

z(x− z)(y− z2 + 2)(y+ z2 − 1),

where the latter two factors are quadrics.

A Groebner base for the SL(2,C) character variety of π1(Ẽ6) is

κ4(x, y, z)(x− y)(xy− z+ 1)(x2 + xy+ y2 − 3) f1(x, y, z) f2(x, y, z),

where κ4(x, y, z) is the SL(2,C) character variety for the fundamental group of the Hopf

link complement, f1(x, y, z) = xy3 − y2z− x2 − 2xy + z + 2 and f2(x, y, z) = y4 − x2z +

xy− 4y2 + z+ 2. A plot of the latter surfaces is in ([8], Figure 4). In the three-dimensional

projective space, the two surfaces are birationally equivalent to a conic bundle and to the

projective plane P2, respectively. Both show a Kodaira dimension zero characteristic of

K3 surfaces.

A Groebner base for the SL(2,C) character variety of π1(D̃4) contains the five-

dimensional hypersurface

f (x, y, z, w, k) = κ4(x, y, z)− wxk− 2k2,

which is close to (but different from) the Fricke form V0,0,w,k(C) = κ4(x, y, z)−wxk+w2 + k2.

Finally, for π1(BR0), a Groebner base obtained from Magma contains 28 polynomials.

However, a simpler small basis with 10 polynomials, like the size of I, is available. The

ideal ring for π1(BR0) is

I = {36 fBR0(x, y, z, u, v, w, k),

xκ4(x, y, z), yκ4(x, y, z), xκ4(x, u, v), yκ4(y, u, v),

−xyk+ xv+ yw+ zk− 2u, xu2 − uzw+ yz− uv+ wk− 2x, .}

(6)

where the seventh variable polynomial reads

fBR0(x, y, z, u, v, w, k) = −xyz+ x

2 + y2 + z2 + xyuk− θ1x− θ2y− θ03z+ θ04

and θ1 = uv+ wk, θ2 = uw+ vk, θ03 = uk− vw, θ04 = u2 + v2 + w2 + k2 − 4.

Taking the new variable z′ = −z+ k, the polynomial fBR0 transforms into the Fricke

form (4).

Vu,v,w,k(x, y, z) = xyz+ x2 + y2 + z2 − θ1x− θ2x− θ3z+ θ4,

(7)

with θ3 = uk+ vw and θ4 = θ04 + uvwk.

The missing term in (6) is a fifth-order polynomial.

4.4. Painlevé VI and the Riemann-Hilbert Correspondence

Equation (7) corresponds to a four-punctured sphere with four singular points and a

monodromy group π1 isomorphic to the free group on three-generators. The existence of a

certain class of linear differential equations with such singular points and a monodromy

group is known as Hilbert’s twenty first problem, the original setting of Riemann–Hilbert

correspondence. For the present case of the four-punctured sphere, the searched differential

(dynamical) equation is the sixth Painlevé equation (or Painlevé VI) [23]

qtt = 12 (

1

q +

1

q−1 +

1

q−t )q

2

t − ( 1t +

1

t−1 +

1

q−t )qt

+ q(q−1)(q−t)

2t2(t−1)2 {α

2

4 − α21

t

q2

+ α22

t−1

(q−1)2 + (1− α

2

3)

t(t−1)

(q−t)2 }

(8)

with complex parameters α1, α2, α3, α4. The Painlevé property is the absence of movable

singular points. The essential singularities of all solutions q(t) of Equation (8) only appear

when t ∈ {0, 1,∞}.

Quantum Rep. 2022, 4

530

Analyzing the nonlinear monodromy of Painlevé VI leads to the relation between

parameters a, b, c and d of the family of cubic surfaces Va,b,c,d(x, y, z) given in (7) and

parameters αi, i = 1 . . . 4 of Painlevé VI equation ([29], Section 4.2):

(a, b, c, d) = [2 cos(πα1), 2 cos(πα2), 2 cos(πα3),−2 cos(πα4)].

(9)

The relation between the two classes of parameters has been found to be controlled

by the so-called Okamoto–Painlevé pairs. The Painlevé equation corresponding to Ẽ8 is

Painlevé I, the Painlevé equation corresponding to Ẽ7 is Painlevé II, the Painlevé equation

corresponding to Ẽ6 is Painlevé IV and the Painlevé equation corresponding to D̃4 is

Painlevé VI ([33] Table 1, [23] Section 9.1.2). These mathematical results fit our approach

developed in the previous subsection.

Incidentally, the Painlevé equation corresponding to the manifold D̃5 is Painlevé V.

We find that the Groebner base for the SL(2,C) character variety of π1(D̃5) contains the

surface κd(x, y, z) defined in Equation (2) (apart from trivial quadratic factors).

Finally, Painlevé III corresponds to one of the three types D̃6, D̃7 or D̃8. We find

that, for Ẽ7 and D̃7, the character variety is trivial (up to quadratic factors), for D̃6, it is

of type κd(x, y, z) and, for D̃8, it is close (but different from the form V0,0,c,d(C), as for D̃4

investigated in Section 4.3.

The Okamoto–Painlevé correspondence and the type of main factor in the related

Groebner base is summarized in Table 1.

Table 1. The manifold type according to the Dynkin diagram (row 1 ), the corresponding Painlevé

equation (row 2) and the main factor in the Groebner base for the corresponding SL(2,C) variety.

The symbol T means that the variety is trivial (up to quadratic factors).

manifold

Ẽ8

Ẽ7

D̃8

D̃7

D̃6

Ẽ6

D̃5

D̃4

Painlevé type

PI

PI I

PD̃8

I I I

PD̃7

I I I

PD̃6

I I I

PIV

PV

PVI

char var

T

T

≈Vo,o,c,d

T

κd

κ4

κd

≈Vo,o,c,d

5. Discussion and Conclusions

In this paper, using the SL(2,C) character variety of the punctured torus S1,1 and

of the quadruply punctured sphere S4,2, we focused on the interest in defining topolog-

ical qubits from the cubic surface κd(x, y, z) in (2) or Va,b,c,d(x, y, z) in (4) in the compact

bounded domain of real variables x, y and z. We explored the connection of such real

surfaces to the character variety of some two- and three-bridge links. We pointed out their

relationship to Painlevé VI transcendents through Okamoto Equation (9). While possible

experimental directions remain open for further investigation, recent advances in the field

are noteworthy [34–37].

Let us now add that there exists a link between Painlevé transcendents and Ein-

stein’s equations of cosmology when the metric is chosen to be self-dual. The six Painlevé

equations are ‘essentially’ equivalent to SL(2,C) self-dual Yang–Mills equations with ap-

propriate three-dimensional Abelian groups of conformal symmetries [38]. The symmetry

groups are taken to be groups of conformal transformations of the complex Minkowski

space–time with the metric

ds2 = dτdτ̄ − dξdξ̄.

For Painlevé VI, the Higgs fields Pi = Φ(Xi), i = 0, 1, t are sl(2,C) valued functions

of the time variable t = ξξ̄

ττ̄ . The self-dual equations

S′ = 0, tP′0 + [P0, Pt], (t− 1)P′1 + [P1, Pt] = 0,

with S = −(P0 + P1 + Pt) are equivalent to Painlevé VI with parameters calculated from

the constant determinants of the Pi and S ([38], p. 573). As a result, the Fricke sur-

faces that we investigated in this paper correspond to relevant solutions of self-dual

Einstein’s equations.

Quantum Rep. 2022, 4

531

Author Contributions: Conceptualization, M.P.; methodology, M.P. and M.M.A.; software, M.P.;

validation, D.C., M.M.A. and K.I.; formal analysis, M.P. and D.C.; investigation, M.M.A., D.C. and

M.M.A.; data curation, M.P.; writing—original draft preparation, M.P.; writing—review and editing,

M.P. and M.M.A.; visualization, M.M.A.; supervision, M.P.; project administration, K.I.; funding

acquisition, K.I. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Informed Consent Statement: Not applicable.

Data Availability Statement: Data are available from the authors after a reasonable demand.

Conflicts of Interest: The authors declare no conflict of interest.

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