Designing Knotted Fields in Light and Electromagnetism

These lectures survey the subject of knotted fields applied to linear waves and optical fields such as laser light. Starting with a description of the topology of complex-valued scalar fields of three dimensional space, and in particular their nodal lines (phase singularities, vortices), the story builds through several different optical settings (random waves, holographically controlled laser light, magnetostatics, time-dependent electromagnetic fields), illustrating different ways knotted fields may be designed and realised physically (via random search, design through braid algebra, geometry of solid angle, and physical intuition). The review is supported by descriptions of the historical development of knotted fields in the 1800s, particularly the work of Helmholtz, Tait and Maxwell, many of whose ideas resonate with recent results.