  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.
