On The Faithfulness of the extension of Lawrence-Krammer representation of the group of conjugating automorphisms C_3

Let \(C_n\) be the group of conjugating automorphisms. We study the representation \(\rho\) of \(C_n\), an extension of Lawrence-Krammer representation of the braid group \(B_n\), defined by Valerij G. Bardakov. As Bardakov proved that the representation \(\rho\) is unfaithful for \(n \geq 5\), the cases \(n=3,4\) remain open. In our work, we make attempts towards the faithfulness of \(\rho\) in the case \(n=3\).