Nuclear Theory

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Lore & Background

Braid groups were introduced explicitly by Emil Artin in 1925, although they were already implicit in Adolf Hurwitz's work on monodromy from 1891. Braid groups may be described by explicit presentations, as shown by Artin in 1947. They are also understood as the fundamental group of certain configuration spaces. Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space, an interpretation rediscovered by Ralph Fox and Lee Neuwirth in 1962. Joan Birman’s book Braids, Links, and Mapping Class Groups (1974) was the first book devoted to braid groups.

In Their Own Story

Tim stands amidst the ruins of a burning world, his hands trembling as he holds the glowing figure close. The light burns through his fingers, a warmth that feels more like pain than comfort. He remembers the pages he found, the warnings about fire and monsters that followed him across every level. Now, with the Princess secured in his arms, the silence is louder than the explosions that never quite ceased.

Reader's Guide

Braid groups have applications in knot theory, where any knot may be represented as the closure of certain braids (Alexander's theorem); in mathematical physics, where Artin's canonical presentation corresponds to the Yang–Baxter equation; and in monodromy invariants of algebraic geometry. Braid theory has also been applied to fluid mechanics, specifically to chaotic mixing in fluid flows, and to the theory of anyons in quantum physics for error-corrected quantum computing.

Did You Know?

• The braid group B_2 is the infinite cyclic group.
• B_3 is isomorphic to the knot group of the trefoil knot.
• All non-identity elements of B_n have infinite order.
• There is a homomorphism from B_n to the symmetric group S_n defined by sending each generator σ_i to the transposition (i, i+1).