  Scenarios in which the knee of the cosmic ray spectrum depends on the
particle rigidities usually predict that the cosmic ray composition becomes
heavier above the knee and have associated a change in the spectral slope of
each individual nuclear component which is steeper than the change
($\Delta\alpha\simeq 0.3$) observed in the total spectrum. We show that this
implies that the very high energy ($E_\nu>10^{14}$ eV) diffuse neutrino fluxes
produced by cosmic rays hitting the atmosphere or colliding with the
interstellar medium in the Galaxy will be significantly suppressed, making
their detection harder but also reducing the background for the search of other
(more challenging) astrophysical neutrino sources.
