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 nstrand braid we denote the crossings of the strands s_{i} and s_{i+1}, when s_{i} overcrosses s_{i+1} by b_{i}, and by b_{i}^{1} when s_{i+1} overcrosses s_{i} (i = 1,..., n1), 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 b_{i}b_{i}^{1} = b_{i}^{1}b_{i} = 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 b_{1}, b_{2}, b_{3},..., and a, b, c, ... instead of b_{1}^{1}, b_{2}^{1}, b_{3}^{1},..., and a^{p} as a shorter expression replacing a...a, where a letter a occurs p times. For example, the short expression ABCd^{2}C replaces b_{1}b_{2}b_{3}b_{4}^{1}b_{4}^{1}b_{3}. In this way, braid words are introduced and isotopies of open braids are described in the language of algebra.
