Copyright © 2010 Mohammad N. Abdulrahim and Madline Al-Tahan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.