  This paper continues the investigation of the relation between the geometry
of a circle embedding and the values of its Fourier coefficients. First, we
answer a question of Kovalev and Yang concerning the support of the Fourier
transform of a starlike embedding. An important special case of circle
embeddings are homeomorphisms of the circle onto itself. Under a one-sided
bound on the Fourier support, such homeomorphisms are rational functions
related to Blaschke products. We study the structure of rational circle
homeomorphisms and show that they form a connected set in the uniform topology.
