1.12  Braids 

K.F. Gauss was the first to notice that braids can be used to describe knotting phenomena. J.W.H. Alexander in 1923 discovered a remarkable connection between KLs and braids. Mathematically speaking, a braid is a formal abstraction of what is meant by a braid in everyday language- some strings tangled in a certain way. An n-strand braid consists of n disjoint arcs running vertically in ³ space, where the starting points lie on the same line. The set of starting points for the arcs must lie immediately above the set of end-points. In a similar way as with KLs, the strands of a braid can be rearranged (without detaching the top and the bottom, and of course without tearing or reattaching them) to get a braid that looks different but is equivalent (or isotopic) to the original braid. Two braids are isotopic if there is a smooth deformation with fixed point from the first one to the second one. As in the case of KLs, the
piecewise-linear category and smooth category give the same result, but further in the sequel we will use piecewise-linear drawings. As with KLs, we do not distinguish between two isotopic braids, thinking of them as two representations of the same object. That is, we are dealing again with an equivalence class of objects. Braid theory was introduced by E. Artin in 1920. That theory connects different fields of mathematics: topology, geometry, algebra (group theory), and algorithmic methods. We obtain the closure of a braid by joining the upper ends of the strands by arcs to the lower ends. J.W.H. Alexander in 1923 proved that every KL can be represented as a closed braid. 

A product of two braids consists of placing two n-strand braids end to end, by joining the upper part of the second braid to the lower part of the first. The product b₁×b₂ of any two n-strand braids b₁ and b₂ is a new n-strand braid (closedness), each three n-strand braids satisfy the relation (b₁×b₂)×b₃ = b₁×(b₂×b₃) (associativity), for every n there is an n-strand unit braid e that does not change a braid that it multiplies (the existence of a neutral element), and for each n-strand braid b there exists an inverse n-strand braid b^-1 whose product with b gives the trivial (unit) n-strand braid e. Let us notice that the braid diagrams of a braid b and its inverse braid b^-1 are mirror-symmetrical in a mirror line containing the joint end of their product b×b^-1. Hence, n-strand braids make a group called a braid group and denoted by (B_n,×). The group (B_n,×) is not commutative: the product of two braids generally depends on the order of the factors.