Because a composite knot complement contain an incompressible torus, the moduli space of flat SU(2) connections modulo gauge contains singularities and parts that are larger than 1-dimensional. For generic perturbations of the flatness equation, the perturbed flat moduli space is 1-dimensional. We describe a way of understanding the topology of these perturbed flat moduli spaces and their images (under restriction) in the flat moduli space of the boundary torus. This, in turn, gives information about the Floer homology of surgeries on these knots.