L-cut splitting of translation surfaces and non-embedding of pseudo-Anosovs (in genus two)

We introduce a concept of a pair of parallel L-cuts on a translation surface, conjecture existence of such pairs for surfaces of genus g > 1, and find them for g = 2. We discuss applications to genus reducing decomposition of surfaces and to pseudo-Anosov maps (concerning their abelian-Nielsen equivalence classes and non-embedding into toral automorphisms). In particular, we provide a negative answer to the question about injectivity of the Abel-Franks map for genus two pseudo-Anosovs with orientable foliations.