  There are large isotope effects in the phonon kinks observed in photoemission
spectra (ARPES) of optimally doped cuprate high temperature superconductors
(HTSC), but they are quite different (Gweon et al. 2004) from those expected
for a nearly free electron metal with strong electron-phonon interactions (Tang
et al. 2003). These differences, together with many other anomalies in infrared
spectra, seem to suggest that other particles (such as magnons) must be
contributing to HTSC. Here we use topological (non-Hamiltonian) methods to
discuss the data, emphasizing nanoscale phase separation and the importance of
a narrow band of quantum percolative states near the Fermi energy that is
spatially pinned to a self-organized filamentary dopant array. Topological
discrete, noncontinuum, nonperturbative methods have previously explained the
form of HTSC phase diagrams without involving detailed microscopic assumptions,
and they are especially useful in the presence of strong nanoscale disorder.
These methods also explain the ``miracle'' of an ideal nearly free electron
phonon kink in sharply defined nodal quasiparticle states in LSCO at the
metal-insulator transition. Finally the universality of the kink energy and
Fermi velocity in different cuprates is the result of the marginally elastic
nature of these materials, and specifically the isostatic character of the CuO2
planes.
