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Krammer's Representation of the Pure Braid Group, P3

We consider Krammer's representation of the pure braid group on three strings: 𝑃3→𝐺𝐿(3,𝑍[𝑡±1,𝑞±1]), where 𝑡 and 𝑞 are indeterminates. As it was done in the case of the braid group, 𝐵3, we specialize the indeterminates 𝑡 and 𝑞 to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, 𝑃3.