# Fricke Topological Qubits

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## Abstract

**:**

## 1. Introduction

## 2. The $\mathit{SL}(\mathbf{2},\mathbb{C})$-Character Variety of a Manifold $\mathbf{M}$

#### A Sage Program for Computing the $SL(2,\mathbb{C})$-Character Variety of ${\pi}_{1}\left(S\right)$

## 3. The Cubic Surface ${\mathit{\kappa}}_{\mathit{d}}(\mathit{x},\mathit{y},\mathit{z})$ and Two-Bridge Links

#### 3.1. The $SL(2,\mathbb{C})$-Character Variety for a Once Punctured Torus

#### 3.2. The Surface ${\kappa}_{2}$, the Link $L7a4={7}_{3}^{2}$ and Fricke Topological Qubits

#### Topological Qubits from ${\kappa}_{2}(x,y,z)$

#### 3.3. The Surface ${\kappa}_{3}(x;y,z)$ and the Link ${6}_{3}^{2}=L6a1$

## 4. The Fricke Cubic Surface and Three-Bridge Links

#### 4.1. The $SL(2,\mathbb{C})$-Character Variety for the Quadruply Punctured Sphere ${S}_{4,2}$

#### 4.2. A Compact Component of $SL(2,\mathbb{R})$

#### 4.3. The $SL(2,\mathbb{C})$ Character Variety for the Manifold $\tilde{{E}_{8}}$ and for Its Covering Manifolds

#### 4.4. Painlevé VI and the Riemann-Hilbert Correspondence

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**

**Top left**: the Cayley cubic ${\kappa}_{4}(x,y,z)$,

**Top right**: the surface ${\kappa}_{2}(x,y,z)$,

**Down**, the surface ${\kappa}_{d}(x,y,z)$ with $d=-\frac{1}{16}$.

**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 ${\kappa}_{2}(x,y,z)$ and ${\kappa}_{4}(x,y,z)$ for the former case, and ${\kappa}_{3}(x,y,z)$ for the latter case.

**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 $SL(2,\mathbb{C})$ variety. The symbol T means that the variety is trivial (up to quadratic factors).

manifold | ${\tilde{E}}_{8}$ | ${\tilde{E}}_{7}$ | ${\tilde{D}}_{8}$ | ${\tilde{D}}_{7}$ | ${\tilde{D}}_{6}$ | ${\tilde{E}}_{6}$ | ${\tilde{D}}_{5}$ | ${\tilde{D}}_{4}$ |

Painlevé type | ${P}_{I}$ | ${P}_{II}$ | ${P}_{III}^{{\tilde{D}}_{8}}$ | ${P}_{III}^{{\tilde{D}}_{7}}$ | ${P}_{III}^{{\tilde{D}}_{6}}$ | ${P}_{IV}$ | ${P}_{V}$ | ${P}_{VI}$ |

char var | T | T | ≈${V}_{o,o,c,d}$ | T | ${\kappa}_{d}$ | ${\kappa}_{4}$ | ${\kappa}_{d}$ | ≈${V}_{o,o,c,d}$ |

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**MDPI and ACS Style**

Planat, M.; Chester, D.; Amaral, M.M.; Irwin, K.
Fricke Topological Qubits. *Quantum Rep.* **2022**, *4*, 523-532.
https://doi.org/10.3390/quantum4040037

**AMA Style**

Planat M, Chester D, Amaral MM, Irwin K.
Fricke Topological Qubits. *Quantum Reports*. 2022; 4(4):523-532.
https://doi.org/10.3390/quantum4040037

**Chicago/Turabian Style**

Planat, Michel, David Chester, Marcelo M. Amaral, and Klee Irwin.
2022. "Fricke Topological Qubits" *Quantum Reports* 4, no. 4: 523-532.
https://doi.org/10.3390/quantum4040037