# Quantum Computing, Seifert Surfaces, and Singular Fibers

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## Abstract

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## 1. Introduction

#### 1.1. Motivation of the Work

#### 1.2. Contents of the Work

## 2. Seifert Surfaces and Braids from d-Fold Coverings of the Trefoil Knot Manifold (or of Hyperbolic Three-Manifolds)

#### 2.1. The Braids Built from the Trefoil Knot that Are Associated with the Qutrit Link $L7n1$ and the Two-Qubit Link $L6a3$

#### 2.2. The Braid Built from the Trefoil Knot that Is Associated with the 6-dit Link ${6}_{3}^{3}$ and Related Braids with the Same Fundamental Group

#### The Six-Cover of the Trefoil Knot Manifold Corresponding to the Congruence Subgroup $3{C}^{0}$ of $\mathsf{\Gamma}$

#### 2.3. The Braid Built from the Trefoil Knot that Is Associated with the Two-Qubit/Qutrit MIC with Icosahedral Symmetry of the Permutation Representation

#### 2.4. Braids from d-Fold Coverings of Hyperbolic Three-Manifolds

#### 2.4.1. The Hyperbolic Link $L10n46$ and Its Zero-Surgery

#### 2.4.2. Further Results

## 3. Quantum Computing from Affine Dynkin Diagrams

#### Reidemeister Torsion of the Manifold ${\mathsf{\Sigma}}^{\prime}$

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The Seifert surface F for a trefoil knot K and (

**b**) the types of crossings for the skein relation of a link L.

**Figure 2.**(

**a**) The link $L7n1$ defining the qutrit MIC, (

**b**) the braid representation, and (

**c**) the corresponding Seifert surface. (

**d**) The link $L6a3$ defining the two-qubit MIC, (

**e**) the braid representation, and (

**f**) the Seifert surface.

**Figure 3.**(

**a**) The link ${6}_{3}^{3}$ corresponding to the 6-dit MIC and the congruence subgroup $\mathsf{\Gamma}\left(2\right)$ of $\mathsf{\Gamma}$, (

**b**) the braid representation, and (

**c**) the Seifert surface. (

**d**) The Kirby link ${L}_{K}$ (see the arrows for the up/down changes), (

**e**) the braid representation, and (

**f**) the Seifert surface (observe the color changes).

**Figure 4.**(

**a**) The Dynkin diagram for the ${D}_{4}$ manifold attached to the 2QB-QT MIC, (

**b**) the corresponding braid $ABCCbaCCBCCb$, and (

**c**) the Seifert surface.

**Figure 5.**A few singular fibers in Kodaira’s classification of minimal elliptic surfaces. (

**a**) Fiber ${I}_{0}^{*}$ (alias ${\tilde{D}}_{4}$), (

**b**) fiber $I{V}^{*}$ (alias ${\tilde{E}}_{6}$), and (

**c**) fiber $I{I}^{*}$ (alias ${\tilde{E}}_{8}$).

**Table 1.**A few models of universal quantum computation (UQC) [7,25] translated into the language of braids and their Seifert surfaces. The source is a knot (such as the trefoil knot) or a link, and the target is a link L associated with a degree d covering of the L-manifold that defines an appropriate magic state for UQC and a corresponding minimal informationally-complete (MIC) measure. Cases $d=3,4,5,\dots $ correspond to the Hesse configuration, to the generalized quadrangle of order two $GQ(2,2)$ (also called a doily), to the Petersen graph. The notation for the braids is that of [20]. The notation ${t}^{\prime}$ means ${t}^{-1}$.

Source | Target | MIC | Braid Word | Alexander Polynomial |
---|---|---|---|---|

trefoil | L7n1 | QTHesse | ${\left(ab\right)}^{3}b$ | ${t}^{5/2}-{t}^{3/2}+{t}^{\prime (3/2)}-{t}^{\prime (5/2)}$ |

. | L6a3 | 2QBdoily | $ABCDCbaCdEdCBCDCeb$ | $-3{t}^{1/2}+3{t}^{\prime (1/2)}$ |

. | ${6}_{3}^{3}$ | 6-ditMIC | ${\left(ab\right)}^{3}$ | ${t}^{2}-t-{t}^{\prime}+{t}^{2}$ |

. | ${D}_{4}$ Dynkin | 2QB-QT MIC | $ABCCbaCCBCCb$ | $-{t}^{3/2}+3{t}^{1/2}+{t}^{\prime (3/2)}-3{t}^{\prime (1/2)}$ |

fig. eight | L10n46 | 2QB doily | $abCbabbcBc$ | $-{t}^{5/2}+4{t}^{3/2}-4{t}^{1/2}+\cdots $ |

. | L14n55217 | 7-dit MIC | $AbbcbcbDacBacdcb$ | $-{t}^{4}+7{t}^{3}-11{t}^{2}+8t-6+\cdots $ |

Whitehead | L12n1741 | QT Hesse | AbcDEFeDCBDacBdcdEdfCbdCddddeD | $-2{t}^{3}+6{t}^{2}-6t+4+\cdots $ |

. | L13n11257 | 5-dit MIC | AbCCbDaCBcDcDcbCD | ${t}^{9/2}-6{t}^{7/2}+15{t}^{5/2}-21{t}^{3/2}$ |

$+21{t}^{1/2}+\cdots $ | ||||

${6}_{3}^{2}=L6a1$ | L12n2181 | QT Hesse | ABcdEFceGbdFaedCBcdEdfcEgbdfedc | $4{t}^{5/2}-12{t}^{3/2}+16{t}^{1/2}+\cdots $ |

. | L14n63905 | 2QB doily | AbCddEdFedcBdaEdfCbceDccDcBC | ${t}^{4}-7{t}^{3}+22{t}^{2}-41t+50+\cdots $ |

L6a5 | L14n63788 | QT Hesse | ABCdEEEFEDcebdacEbEED | ${t}^{4}-2{t}^{3}+2t-2+\cdots $ |

ceDefedCeBdCEDe |

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**MDPI and ACS Style**

Planat, M.; Aschheim, R.; Amaral, M.M.; Irwin, K.
Quantum Computing, Seifert Surfaces, and Singular Fibers. *Quantum Rep.* **2019**, *1*, 12-22.
https://doi.org/10.3390/quantum1010003

**AMA Style**

Planat M, Aschheim R, Amaral MM, Irwin K.
Quantum Computing, Seifert Surfaces, and Singular Fibers. *Quantum Reports*. 2019; 1(1):12-22.
https://doi.org/10.3390/quantum1010003

**Chicago/Turabian Style**

Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin.
2019. "Quantum Computing, Seifert Surfaces, and Singular Fibers" *Quantum Reports* 1, no. 1: 12-22.
https://doi.org/10.3390/quantum1010003