Burau representation

The unreduced Burau representation has a similar definition, namely one replaces Dn with its (real, oriented) blow-up at the marked points.

Then the unreduced Burau representation may be given explicitly by mapping for 1 ≤ i ≤ n − 1, where Ik denotes the k × k identity matrix.

The first nonfaithful Burau representations were found by John A. Moody without the use of computer, using a notion of winding number or contour integration.

[3] A more conceptual understanding, due to Darren D. Long and Mark Paton[4] interprets the linking or winding as coming from Poincaré duality in first homology relative to the basepoint of a covering space, and uses the intersection form (traditionally called Squier's Form as Craig Squier was the first to explore its properties).

[5] Stephen Bigelow combined computer techniques and the Long–Paton theorem to show that the Burau representation is not faithful for n ≥ 5.

[5] Moreover, when the variable t is chosen to be a transcendental unit complex number near 1, it is a positive-definite Hermitian pairing.

The covering space C n may be thought of concretely as follows: cut the disk along lines from the boundary to the marked points. Take as many copies of the result as there are integers, stack them vertically, and connect them by ramps going from one side of the cut on one level to the other side of the cut on the level below. This procedure is shown here for n = 4 ; the covering transformations t ±1 act by shifting the space vertically.