  Exciton Scattering (ES) theory attributes excited electronic states to
standing waves in quasi-one-dimensional molecular materials by assuming a
quasi-particle picture of optical excitations. The quasi-particle properties at
branching centers are described by the corresponding scattering matrices. Here
we identify the topological invariant of scattering center, referred to as its
winding number, and apply topological intersection theory to count the number
of quantum states in a quasi-one-dimensional system.
