On the Milnor fiber boundary of a quasi-ordinary surface

We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. We indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.