  We review the q-deformed spin network approach to Topological Quantum Field
Theory and apply these methods to produce unitary representations of the braid
groups that are dense in the unitary groups. Our methods are rooted in the
bracket state sum model for the Jones polynomial. We give our results for a
large class of representations based on values for the bracket polynomial that
are roots of unity. We make a separate and self-contained study of the quantum
universal Fibonacci model in this framework. We apply our results to give
quantum algorithms for the computation of the colored Jones polynomials for
knots and links, and the Witten-Reshetikhin-Turaev invariant of three
manifolds.
