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 (Δα≃0.3) observed in the total spectrum. We show that this implies that the very high energy (Eν>1014 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.Facultad de Ciencias Exactas