Forks, noodles and the Burau representation for n=4.

The reduced Burau representation is a natural action of the braid group on the first homology group of a suitable infinite cyclic covering space of the 4-punctured disc . It is known that the Burau representation is faithful for n = 1,2,3 and that it is not faithful for n > 4. 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.