Michel Planat, Raymond Aschheim, Marcelo M. Amaral, Klee Irwin (2019)

The fundamental group π1(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 E6 and D4, 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 Σ′, at the boundary of the singular fiber E8~, allows possible models of quantum computing.