Let Bn denote the n-strand braid group. We recall that this admits a representation βn:Bn→GLn-1(Z[t,t-1]) the (reduced) Burau representation [1]. Although open for a long time, it is now known that this representation is not faithful for n≥6, (See [8,5]) and it is an old result that it is faithful for n=3. (See [7]) Despite these counterexamples, there is no understanding of the nature of the image groups in the nonfaithful cases nor any kind of intrinsic characterisation of braids which lie in the kernel. The two cases n=4,5 remain open. Resolution of the case n=4 is an important open problem, firstly for the implications for the automorphism group of a free group of rank 2 and secondly as a test case for the faithfulness of the Jones representation, [4]. In the case n=4 the only summand.
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机译：令Bn表示n股辫子群。我们记得，这承认了一个表示（减少的）Burau表示[1]：βn：Bn→GLn-1（Z [t，t-1]）。尽管开放很长时间，但现在知道该表示对于n≥6不忠实（请参阅[8,5]），并且对于n = 3是忠实的是一个古老的结果。 （见[7]）尽管有这些反例，但对于不忠实的情况下的图像组的性质也没有任何了解，也没有对位于内核中的辫子的任何内在表征。 n = 4,5的两种情况仍未解决。 n = 4的情况的解决是一个重要的开放问题，首先是对自由等级2的自同构群的影响，其次是对Jones表示的真实性的测试用例，[4]。在n = 4的情况下，唯一求和。
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