  In previous work with Schoenfeld, we considered a string-type chain complex
of curves on surfaces, with differential given by resolving crossings, and
computed the homology of this complex for discs.
  In this paper we consider the corresponding "string homology" of annuli. We
find this homology has a rich algebraic structure which can be described, in
various senses, as fermionic. While for discs we found an isomorphism between
string homology and the sutured Floer homology of a related 3-manifold, in the
case of annuli we find the relationship is more complex, with string homology
containing further higher-order structure.
