#
$SL(2,\mathbb{C})$ Scheme Processing of Singularities in Quantum Computing and Genetics

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## Abstract

**:**

_{4}; topological quantum computing; microRNAs

## 1. Introduction

## 2. Theory

#### 2.1. The $SL(2,\mathbb{C})$ Character Variety of a Finitely Generated Group and a Groebner Basis

#### 2.2. Singularities of an Algebraic Surface

#### Simple Singularities

#### 2.3. Arbitrary Singularities

#### 2.4. Kodaira–Enriques Classification

#### A Singular Surface

## 3. $\mathbf{SL}(\mathbf{2},\mathbb{C})$ Scheme Processing in Quantum Computing Based on an Akbulut Cork

#### 3.1. A Short Account of Magic States for Quantum Computing

#### 3.2. Brief Introduction to 4-Manifolds

#### 3.3. Akbulut Cork

#### 3.4. The Manifold $\overline{W}$ Mediating the Akbulut Cobordism between Exotic Manifolds V and W

#### 3.5. The Character Variety for an Akbulut Cork W

#### Formal Desingularization of the Surface ${S}_{W}(x,y,z)$

#### 3.6. The Character Variety for the Mediating Manifold $\overline{W}$

## 4. $\mathit{SL}(\mathbf{2},\mathbb{C})$ Scheme Processing in Topological Quantum Computing

## 5. $\mathit{SL}(\mathit{2},\mathbb{C})$ Scheme Processing in microRNAs

## 6. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Planat, M.; Minarovjech, M.; Saniga, M. Ramanujan sums analysis of long-period sequences and 1/f noise. EPL
**2009**, 85, 40005. [Google Scholar] [CrossRef][Green Version] - Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys.
**1976**, 48, 393–416. [Google Scholar] [CrossRef][Green Version] - Rovelli, C.; Vidotto, F. Covariant Loop Quantum Gravity, 1st ed.; Cambridge University Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Character Variety. Available online: https://en.wikipedia.org/wiki/Character_variety (accessed on 1 January 2022).
- Bullock, D. Rings of SL
_{2}(C)-characters and the Kauffman bracket skein module. CM Helv.**1997**, 72, 521–542. [Google Scholar] - Gröbner Basis. Available online: https://en.wikipedia.org/wiki/Gröbner_basis (accessed on 1 August 2022).
- ADE Singularity. Available online: https://ncatlab.org/nlab/show/ADE+singularity (accessed on 1 August 2022).
- Reid, M. Undergraduate Commutative Algebra; London Mathematical Society Student Texts; Cambridge University Press: Cambridge, MA, USA, 1995; Volume 29. [Google Scholar]
- Hartshorne, R. Algebraic Geometry; Graduate Texts in Mathematics; Springer: New York, NY, USA, 1977; Volume 52. [Google Scholar]
- Scheme (Mathematics). Available online: https://en.wikipedia.org/wiki/Scheme_(mathematics) (accessed on 1 May 2022).
- Planat, M.; Chester, D.; Amaral, M.; Irwin, K. Fricke topological qubits. Quant. Rep.
**2022**, 4, 523–532. [Google Scholar] [CrossRef] - Planat, M.; Amaral, M.M.; Irwin, K. Algebraic morphology of DNA–RNA transcription and regulation. Preprints
**2022**. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Quantum Computation and Measurements from an Exotic Space-Time R
^{4}. Symmetry**2020**, 12, 736. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Group geometrical axioms for magic states of quantum computing. Mathematics
**2019**, 7, 948. [Google Scholar] [CrossRef][Green Version] - Python Code to Compute Character Varieties. Available online: http://math.gmu.edu/~slawton3/Main.sagews (accessed on 1 May 2021).
- Bosma, W.; Cannon, J.J.; Fieker, C.; Steel, A. (Eds.) Handbook of Magma Functions, 2.23 ed.; University of Sydney: Sydney, NSW, Australia, 2017; 5914p. [Google Scholar]
- Goldman, W.M. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. Eur. Math. Soc. Zürich
**2009**, 13, 611–684. [Google Scholar] - Inaba, M.; Iwasaki, K.; Saito, M.H. Dynamics of the sixth Painlevé equation. arXiv
**2005**, arXiv:math.AG/0501007. [Google Scholar] - Beck, T.; Schicho, J. Adjoint computation for hypersurfaces using formal desingularizations. J. Algebra
**2018**, 320, 3984–3996. [Google Scholar] [CrossRef][Green Version] - Schicho, J. Rational parametrization of surfaces. J. Symb. Comput.
**1998**, 26, 1–29. [Google Scholar] [CrossRef][Green Version] - Enriques Kodaira Classification. Available online: https://en.wikipedia.org/wiki/Enriques-Kodaira_classification (accessed on 1 January 2022).
- Bravyi, S.; Kitaev, A. Universal quantum computation with ideal Clifford gates and noisy ancillas. Phys. Rev. A
**2005**, 71, 022316. [Google Scholar] [CrossRef][Green Version] - Veitch, V.; Mousavian, S.A.; Gottesman, D.; Emerson, J. The Resource Theory of Stabilizer Computation. New J. Phys.
**2014**, 16, 013009. [Google Scholar] [CrossRef] - Akbulut, S. 4-Manifolds; Oxford Graduate Texts in Mathematics; Oxford University Press: Oxford, UK, 2016; Volume 25. [Google Scholar]
- Gompf, R.E.; Stipsicz, A.I. 4-Manifolds and Kirby Calculus; Graduate Studies in Mathematics; American Mathematical Society: Providence, Rhode Island, 1999; Volume 20. [Google Scholar]
- Scorpian, A. The Wild World of 4-Manifolds; American Mathematical Society: Providence, Rhode Island, 2011. [Google Scholar]
- Akbulut, S. A fake compact contractible 4-manifold. J. Diff. Geom.
**1991**, 33, 335–356. [Google Scholar] [CrossRef] - Akbulut, S. An exotic 4-manifold. J. Diff. Geom.
**1991**, 33, 357–361. [Google Scholar] [CrossRef] - Akbulut, S.; Durusoy, S. An involution acting nontrivially on Heegard-Floer homology. In Geometry and Topology of Manifolds; Fields Institute Communication, American Mathematical Society: Providence, RI, USA, 2005; Volume 47, pp. 1–9. [Google Scholar]
- Planat, M. Geometry of contextuality from Grothendieck’s coset space. Quantum Inf. Process.
**2015**, 14, 2563–2575. [Google Scholar] [CrossRef] - Medley, C.M.; Panzade, G.; Zinovyeva, A.Y. MicroRNA strand selection: Unwinding the rules. WIREs RNA
**2021**, 12, e1627. [Google Scholar] [CrossRef] - Dawson, O.; Piccinini, A.M. miR-155-3p: Processing by-product or rising star in immunity and cancer? Open Biol.
**2022**, 12, 220070. [Google Scholar] [CrossRef] - Kozomara, A.; Birgaonu, M.; Griffiths-Jones, S. miRBase: From microRNA sequences to function. Nucl. Acids Res.
**2019**, 47, D155–D162. [Google Scholar] [CrossRef] - miRBase: The microRNA Database. Available online: https://www.mirbase.org/ (accessed on 1 November 2022).
- Fromm, B.; Billipp, T.; Peck, L.E.; Johansen, M.; Tarver, J.E.; King, B.L.; Newcomb, J.M.; Sempere, L.F.; Flatmark, K.; Hovig, E.; et al. A uniform system for the annotation of human microRNA genes and the evolution of the human microRNAome. Annu. Rev. Genet.
**2015**, 23, 213–242. [Google Scholar] [CrossRef][Green Version] - He, Y.; Cai, Y.; Paii, P.M.; Ren, X.; Xia, Z. The Causes and Consequences of miR-503 Dysregulation and Its Impact on Cardiovascular Disease and Cancer. Front. Pharmacol.
**2021**, 12, 629611. [Google Scholar] [CrossRef] [PubMed] - Amin, M.M.J.; Trevelyan, C.J.; Tyurner, N.A. MicroRNA-214 in health and disease. Cells
**2021**, 23, 3274. [Google Scholar] [CrossRef] [PubMed] - Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K. Informationally complete characters for quark and lepton mixings. Symmetry
**2020**, 12, 1000. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Irwin, K. Complete quantum information in the DNA genetic code. Symmetry
**2020**, 12, 1993. [Google Scholar] [CrossRef] - Planat, M.; Aschheim, R.; Amaral, M.M.; Fang, F.; Chester, D.; Irwin, K. Character varieties and algebraic surfaces for the topology of quantum computing. Symmetry
**2022**, 14, 915. [Google Scholar] [CrossRef] - Catren, G.; Cukierman, F. Grothendieck’s theory of schemes and the algebra–geometry duality. Synthese
**2022**, 200, 234. [Google Scholar] [CrossRef] - Decker, W.; Greuel, G.M. History of Singular and its relation to Zariski’s multiplicity conjecture. arXiv
**2012**, arXiv:2103.00525. [Google Scholar] - Singular. Available online: https://www.singular.uni-kl.de (accessed on 1 January 2023).
- Asselmeyer-Maluga, T.; Król, J.; Wilms, A. Big bang and topology. Symmetry
**2022**, 14, 1887. [Google Scholar] [CrossRef] - Meijer, D.K.F.; Raggett, S. Quantum Physics in Consciousness Studies. Available online: https://www.deeplook.ir/wp-content/uploads/2016/07/Quantum-Ph-rev-def-2.pdf (accessed on 1 January 2023).

**Figure 1.**The affine singular surface ${S}_{2}(x,y,z)={z}^{4}+2y{z}^{3}+{x}^{2}-6yz-2x-8$ found in the Groebner basis for the transcription factor Prdm1 ([12], Section 3.1).

**Figure 2.**(

**a**) Handlebody of a 4-manifold with the structure of 1- and 2-handles over the 0-handle ${B}^{4}$, (

**b**) the structure of a 1-handle as a dotted circle ${S}^{1}\times {B}^{3}$, and (

**c**) an Akbulut cork $W={9}_{46}(-1,1)$.

**Figure 3.**The surface ${S}_{W}(x,y,z)$ found within the Groebner basis of the $SL(2,\mathbb{C})$ character variety for the Akbulut cork W.

**Figure 4.**Left: the (degree 3) del Pezzo surface ${S}_{1}(x,y,z)={f}_{2,\left\{\right\}}^{\left({A}_{1}\right)}(x,y,z)=xyz+x{z}^{2}+{x}^{2}+{y}^{2}+{z}^{2}+yz\phantom{\rule{4pt}{0ex}}-x-6$. Middle: the (rational scroll) surface ${S}_{2}(x,y,z)=x{z}^{2}+yz-x-2$. Right: the (del Pezzo degree 4) surface ${S}_{3}(x,y,z)=-{x}^{2}{z}^{2}+2{x}^{2}+{y}^{2}+{z}^{2}+2x-4$.

**Figure 5.**Left: the Cayley cubic ${\kappa}_{4}(x,y,z)$ found in the character variety for the slowest evolving miRNA gene hsa-mir-503. The surface ${f}_{1,\{1:0:0:0\}}^{\left(A2\right)}(x,y,z)$ found in the character variety ${\mathcal{G}}_{mir-214-5p}$ of the fast evolving gene hsa-mir-214.

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**MDPI and ACS Style**

Planat, M.; Amaral, M.M.; Chester, D.; Irwin, K.
*Axioms* **2023**, *12*, 233.
https://doi.org/10.3390/axioms12030233

**AMA Style**

Planat M, Amaral MM, Chester D, Irwin K.
*Axioms*. 2023; 12(3):233.
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**Chicago/Turabian Style**

Planat, Michel, Marcelo M. Amaral, David Chester, and Klee Irwin.
2023. "*Axioms* 12, no. 3: 233.
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