  We show that the realisation of synthetic magnetic fields via light-matter
coupling in the Lambda-scheme implements a natural geometrical construction of
magnetic fields, namely as the pullback of the area element of the sphere to
Euclidean space via certain maps. For suitable maps, this construction
generates linked and knotted magnetic fields, and the synthetic realisation
amounts to the identification of the map with the ratio of two Rabi frequencies
which represent the coupling of the internal energy levels of an ultracold
atom. We consider examples of maps which can be physically realised in terms of
Rabi frequencies and which lead to linked and knotted synthetic magnetic fields
acting on the neutral atomic gas. We also show that the ground state of the
Bose-Einstein condensate may inherit topological properties of the synthetic
gauge field, with linked and knotted vortex lines appearing in some cases.
