The Faithfulness Question for the Burau Representation

We consider the Burau and Gassner representations of the braid groups B„ . A sufficient condition for faithfulness, involving just a pair of arcs, is shown to be necessary as well for all but at most two values of n. In the Burau case, this implies nonfaithfulness for n > 10 . Alexander showed in [6] that any knot K can be made by connecting the ends of some braid b. The Alexander polynomial A(K) was defined in terms of a certain matrix A of relations. In 1926, Alexander found that A(K) does not uniquely determine the knot K [1]. Subsequently, Burau wrote down, for each braid b, a matrix of Alexander relations of the form A(b) 1 for the associated knot or link, such that the matrix of a composite bb' satisfies