Braid Groups

Braid groups are developed here from configuration spaces rather than from Artin's presentation, so that the group of a braid is a fundamental group before it is a word. Ordered and unordered configuration spaces of the disc are compared through the free coordinate-permutation action, the induced covering is shown to be regular with deck group the symmetric group, and the resulting short exact sequence isolates the pure braid group. Fadell-Neuwirth forgetful fibrations, punctured discs and point pushing then supply the computational machinery, and the Artin presentation, braid combing and the Garside normal form are recovered from these configuration models. The Artin action on a free group, the Burau representations, and the Lawrence-Krammer-Bigelow linearity theorem follow, together with oriented links, braid closures and Markov equivalence, and the Hecke-Markov traces that produce polynomial link invariants. The categorical half of the collection works over graded quiver algebras and their derived tensor functors, Khovanov-Seidel twist complexes, categorical braid actions, Soergel bimodules, Rouquier complexes, matrix factorizations and Hochschild homology, keeping the decategorified and categorified accounts on separate pages. Sources are cited as they are used, and the Axiom of Choice is declared only where a partition of unity or a well-ordering genuinely enters.