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 nstrand
braid consists of n disjoint arcs running vertically in Â^{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 endpoints.
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 nstrand braids end to end, by joining the upper part of the second braid to the lower part of the first. The product b_{1}×b_{2} of any two nstrand braids b_{1} and b_{2} is a new nstrand braid (closedness), each three nstrand braids satisfy the relation (b_{1}×b_{2})×b_{3} = b_{1}×(b_{2}×b_{3}) (associativity), for every n there is an nstrand unit braid e that does not change a braid that it multiplies (the existence of a neutral element), and for each nstrand braid b there exists an inverse nstrand braid b^{1} whose product with b gives the trivial (unit) nstrand braid e. Let us notice that the braid diagrams of a braid b and its inverse braid b^{1} are mirrorsymmetrical in a mirror line containing the joint end of their product b×b^{1}. Hence, nstrand 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.
