  We study the limiting zero distribution of orthogonal polynomials with
respect to some particular exponential weights exp(-nV(z)) along contours in
the complex plane. We are especially interested in the question under which
circumstances the zeros of the orthogonal polynomials accumulate on a single
analytic arc (one cut case), and in which cases they do not. In a family of
cubic polynomial potentials V(z) = - iz^3/3 + iKz, we determine the precise
values of K for which we have the one cut case. We also prove the one cut case
for a monomial quintic V(z) = - iz^5/5 on a contour that is symmetric in the
imaginary axis.
