On Representations of Braid Groups

To understand what a braid group is, it is easiest to visualize a braid. Consider n strands, all parallel. Consider taking the ith strand and crossing it over the very next strand. This is a braid. In fact, a braid is any sequence of crossings of the bands, provided none of the bands are self-crossing. For instance, a loop, or a band which forms a loop in the middle are not braids. Now, in order for the set of all braids of n bands to be group, we must be able to find a binary operation on the braids that satisfies certain properties. We consider a very simple operation which we call concatenation. We take two braids, and do the sequence of twists of the first one followed by the twists of the second braid. The result is another braid. Thus, this operation is closed. The identity element for this operation is also obvious. Clearly, the bands not having any crossings is the identity, as this braid concatenated with any other braid is just the other braid. For the inverse of x, one just takes the identity element, and do the exact opposite crossings in the exact opposite sequence than the sequence of crossings in x. Thus, The set of all braids on n strands forms a group under concatenation. Artin showed that the braid group permits permutation by the Artin generators, which obey two relations