Where the Braid Is a Proxy

Three registers, kept apart

Mathematics: the braid group and its invariants (Chapter 10). These are theorems. They constrain any system whose configuration space is the braid group and constrain nothing else.

Physical structure: optical orbital angular momentum (Chapter 9), topological invariants and protected edges (Chapter 11), anyonic exchange (Chapter 12), quantised vortex circulation (Chapter 13). Here the braided or topological quantity is measurable, and the integer that appears in an equation is the integer an instrument returns.

Figure: the silk sheet with its threads of braided light (Chapter 7), the resemblance between correlation hypergraphs and neural wiring (Chapter 14), and every use in the wider corpus of 'braided' to mean richly interrelated. These are held for their usefulness in thinking and carry no evidential weight. They are labelled here so they cannot be cashed elsewhere.

The test that sorts them

One question does most of the work: does the crossing change the state? If exchanging two elements in a different order produces a different, measurable outcome, braiding is doing physical work. If the picture would survive being drawn in any order without changing a prediction, the braid is a figure — an aid to memory and attention, not a mechanism. Most uses of the word in popular writing, and some in this author's earlier books, fail that test. Saying so is the point of the chapter.