braid group

《Symplectic Geometry》 · 第 183–184 页 · Section 7.10，圆柱缠结叠合、辫等价及群结构

A cylindrical \(n\)-tangle is a ball \(D^2\times[0,1]\) with a line of \(n\) inputs in the top (\(D^2\times1\)) and a line of \(n\) outputs directly below them at the bottom. […] Each disc \(D^2\times x\) in the cylinder meets the inhabitant in exactly \(n\) points, once in each string; there are no extra loops. Such a tangle is called an \(n\)-string braid. Its strings descend monotonically without reversing anywhere. Braids are equivalent if they are isotopic using braid-preserving isotopies so that all the intervening positions are also braids. […] These tangles have a composition rule: stack two cylinders on top of each other to produce another. […] The \(n\)-string braids form a group (Exercise 7.12.18), denoted by \(B_n\), and this allows some algebraic methods to be used in knot theory.