  Given two irreducible curves of the plane which have isomorphic complements,
it is natural to ask whether there exists an automorphism of the plane that
sends one curve on the other.
  This question has a positive answer for a large family of curves and
H.Yoshihara conjectured that it is true in general. We exhibit counterexamples
to this conjecture, over any ground field. In some of the cases, the curves are
isomorphic and in others not; this provides counterexamples of two different
kinds.
  Finally, we use our construction to find the existence of surprising
non-linear automorphisms of affine surfaces.
