  Little is known on the classification of Heegaard splittings for hyperbolic
3-manifolds. Although Kobayashi gave a complete classification of Heegaard
splittings for the exteriors of 2-bridge knots, our knowledge of other classes
is extremely limited. In particular, there are very few hyperbolic manifolds
that are known to have a unique minimal genus splitting. Here we demonstrate
that an infinite class of hyperbolic knot exteriors, namely exteriors of
certain "twisted torus knots" originally studied by Morimoto, Sakuma and
Yokota, have a unique minimal genus Heegaard splitting of genus two. We also
conjecture that these manifolds possess irreducible yet weakly reducible
splittings of genus three. There are no known examples of such Heegaard
splittings.
