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Invent. math. 127, 535570 (1997) A presentation for the image of Burau(4) Z2

Summary: Invent. math. 127, 535­570 (1997)
A presentation for the image of Burau(4) Z2
D. Cooper?
, D.D. Long??
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
Oblatum 28-XII-1995 & 5-XI-1996
1 Introduction
Let Bn denote the n-strand braid group. We recall that this admits a represen-
˙n : 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, ˙rstly 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


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics