Abstract
The reduced Burau representation is a natural action of the braid group Bn on the first homology group H1(D̃n;Z) of a suitable infinite cyclic covering space D̃n of the n-punctured disc Dn. It is known that the Burau representation is faithful for n≤3 and that it is not faithful for n≥5. We use forks and noodles homological techniques and Bokut–Vesnin generators to analyze the problem for n=4. We present a Conjecture implying faithfulness and a Lemma explaining the implication. We give some arguments suggesting why we expect the Conjecture to be true. Also, we give some geometrically calculated examples and information about data gathered using a C++ program.
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