A braid with a single crossing is called an elementary braid. The pictures representing braids can be algebraically encoded. Moving along a braid from top to bottom in successive levels, we see that the braid can be represented as the successive product of elementary braids. If in an n-strand braid we denote the crossings of the strands si and si+1, when si overcrosses si+1 by bi, and by bi-1 when si+1 overcrosses si (i = 1,..., n-1), we obtain algebraic codes for braids- braid words. Expressed in terms of braid words, the equivalence relation- isotopy of braids is described by the following relations: commutativity for distant braids
Together with trivial relations bibi-1 = bi-1bi = e they are sufficient for replacing geometric manipulations related to isotopy by algebraic calculations consisting in replacing a part of a word by its equivalent word according to word relations mentioned. Thus, two braids are isotopic iff the word representing one of them can be transformed into the word representing the other by a finite series of word replacements satisfying the relations (1.1), (1.2). In order to simplify the notation, we will use A, B, C, ... instead of b1, b2, b3,..., and a, b, c, ... instead of b1-1, b2-1, b3-1,..., and ap as a shorter expression replacing a...a, where a letter a occurs p times. For example, the short expression ABCd2C replaces b1b2b3b4-1b4-1b3. In this way, braid words are introduced and isotopies of open braids are described in the language of algebra.