Projective representations of link groups

This dissertation is a study of the representations of a knot or link group in PSL(2,cal C). We determine all the representations of a 2-bridge knot group in PSL(2,cal C) and find that they correspond to the points of an algebraic plane curve, ϵ. This curve has no multiple components, the points on it corresponding to the parabolic representations of the knot group are simple, and the generic representations for at least one component of ϵ are faithful. Also ϵ avoids a certain region of the complex projective plane. We next relate the projective representations of the group πKo of a knot ko to the parabolic representations of the group πK of a satellite k of k_o. We define a primitive parabolic representation of πK and show that the primitive representations are non-rigid in an algebraic sense. We then consider the special case where k is a Whitehead double of ko in more detail. A primitive parabolic representation is shown to correspond to the solutions of a single algebraic equation whose unknowns concern the projective representations of πK_o. We show that all doubles of torus knots admit parabolic representations. The final topics are certain subgroups G of PSL(2,cal C) that are generated by parabolic transformations but are not free. We show how to associate an infinite sequence of such groups G to every 2-bridge knot or link, and we show that the groups in two such sequences are discrete and have presentations that were predicted by the setup.