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 Â3
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
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 (Bn,×). The group (Bn,×) is not commutative: the product of two braids generally depends on the order of the factors.