Search

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

In this paper, we will present some ideas to use 3D topology for quantum computing extending ideas from a previous paper. Topological quantum computing used "knotted" quantum states of topological phases of matter, called anyons. But anyons are connected with surface topology. But surfaces have (usually) abelian fundamental groups and therefore one needs non-Abelian anyons to use it for quantum computing. But usual materials are 3D objects which can admit more complicated topologies. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere (see T. Asselmeyer-Maluga, Quantum Rep. 3 (2021) 153, arXiv:2102.04452 for previous work). The whole system is designed as knotted superconductor, where every crossing is a Josephson junction and the qubit is realized as flux qubit. We discuss the properties of this systems in particular the fluxion quantization by using the A-polynomial of the knot. Furthermore, we showed that 2-qubit operations can be realized by linked (knotted) superconductors again coupled via a Josephson junction. [ABSTRACT FROM AUTHOR]

It is known that the fundamental group homomorphism π 1 ( T 2 ) → π 1 ( S 3 ∖ K ) induced by the inclusion of the boundary torus into the complement of a knot K in S 3 is a complete knot invariant. Many classical invariants of knots arise from the natural (restriction) map induced by the above homomorphism on the SL 2 -character varieties of the corresponding fundamental groups. In our earlier work [3] , we proposed a conjecture that the classical restriction map admits a canonical deformation into a two-parameter family of affine cubic surfaces in C 3 . In this paper, we show that (modulo some mild technical conditions) our conjecture follows from a known conjecture of Brumfiel and Hilden [1] on the algebraic structure of the peripheral system of a knot. We then confirm the Brumfiel–Hilden conjecture for an infinite class of knots, including all torus knots, 2-bridge knots, and certain pretzel knots. We also show the class of knots for which the Brumfiel–Hilden conjecture holds is closed under taking connect sums and knot coverings. [ABSTRACT FROM AUTHOR]

We find the family of all knots in S3 which are spanned by two essential once-punctured Klein bottles with boundary slopes at distance 4, thus settling a conjecture by K. Ichihara, M. Ohtouge, and M. Teragaito. We also address the more general question of when a knot exterior in an arbitrary 3-manifold contains two essential once-punctured Klein bottles with distinct boundary slopes. [Copyright &y& Elsevier]