On the Faithfulness of the Extension of Lawrence-Krammer Representation of the Group of Conjugating Automorphisms C3

Let Cn be the group of conjugating automorphisms. We study the representation ρ of Cn, an extension of Lawrence-Krammer representation of the braid group Bn. It is known that the representation ρ is unfaithful for n ≥ 5, the cases n = 3, 4 remain open. In our work, we make attempts towards the faithfulness of ρ in the case n = 3.