  This paper contains a preliminary study of the monodromy of certain fourth
order differential equations, that were called of Calabi-Yau type in
math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs
equations of a Calabi-Yau manifold with one complex modulus, which links up the
observed integrality to the conjectured integrality of the Gopakumar-Vafa
invariants. A natural question is if in the other cases such a geometrical
interpretation is also possible. Our investigations of the monodromies are
intended as a first step in answering this question. We use a numerical
approach combined with some ideas from homological mirror symmetry to determine
the monodromy for some further one-parameter models. Furthermore, we present a
conjectural identification of the Picard-Fuchs equation for 5 new examples from
Borcea's list and one constructed by Tonoli and conjecture the existence of
some new Calabi-Yau three folds. The paper does not contain any theorems or
proofs but is, we think, nevertheless of interest.
