  The observations of a nearby low-luminosity gamma-ray burst (GRB) 060218
associated with supernova SN 2006aj may imply an interesting astronomical
picture where a supernova shock breakout locates behind a relativistic GRB jet.
Based on this picture, we study neutrino emission for early afterglows of GRB
060218-like GRBs, where neutrinos are expected to be produced from photopion
interactions in a GRB blast wave that propagates into a dense wind.
Relativistic protons for the interactions are accelerated by an external shock,
while target photons are basically provided by the incoming thermal emission
from the shock breakout and its inverse-Compton scattered component. Because of
a high estimated event rate of low-luminosity GRBs, we would have more
opportunities to detect afterglow neutrinos from a single nearby GRB event of
this type by IceCube. Such a possible detection could provide evidence for the
picture described above.
