On the Linearity of Braid Groups
نویسنده
چکیده
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 i+ 1th strand. This is an example of a braid. In general, a braid is any sequence of crossings of the bands, provided none of the bands are selfcrossing. For instance, a loop, or a band which forms a loop in the middle, is not a braid. 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. To find inverse element of x, one just takes the identity element, and do the exact opposite crossings of x 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 [1] showed that the braid group is generated by the Artin generators, which obey two relations:
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