Another Proof of the Faithfulness of the Lawrence-Krammer Representation of the Braid Group B3

Abstract EN

The Lawrence-Krammer representation of the braid group Bn was proved to be faithful for n≥3 by Bigelow and Krammer.

In our paper, we give a new proof in the case n=3 by using matrix computations.

First, we prove that the representation of the braid group B3 is unitary relative to a positive definite Hermitian form.

Then we show the faithfulness of the representation by specializing the indeterminates q and t to complex numbers on the unit circle rather than specializing them to real numbers as what was done by Krammer.