# Quantum computing, Seifert surfaces and singular fibers

## TL;DR

This paper explores the connection between knot theory, Seifert surfaces, and quantum computing, proposing models based on the topology of links and their coverings to facilitate universal quantum computation.

## Contribution

It introduces a novel approach linking Seifert surfaces and Alexander polynomials of specific knots to quantum computational models.

## Key findings

- Certain coverings of the trefoil knot relate to Dynkin diagrams of E6 and D4.
- The correspondence extends to Kodaira's classification of elliptic singular fibers.
- A Seifert fibered manifold model is proposed for quantum computing applications.

## Abstract

The fundamental group $\pi_1(L)$ of a knot or link $L$ may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the $L$ of such a quantum computer model and computes their Seifert surfaces and the corresponding Alexander polynomial. In particular, some $d$-fold coverings of the trefoil knot, with $d=3$, $4$, $6$ or $12$, define appropriate links $L$ and the latter two cases connect to the Dynkin diagrams of $E_6$ and $D_4$, respectively. In this new context, one finds that this correspondence continues with the Kodaira's classification of elliptic singular fibers. The Seifert fibered toroidal manifold $\Sigma'$, at the boundary of the singular fiber $\tilde {E_8}$, allows possible models of quantum computing.