  A recent proposal to associate 60~TeV to 2~PeV IceCube neutrino events with
gamma-ray bursts~(GRBs) indicates the Lorentz violation of cosmic neutrinos and
leads further to the $CPT$ symmetry violation between neutrinos and
antineutrinos. Here we find that another 12 northern hemisphere track events
possibly correlated with GRBs from three-year IceCube data satisfy the same
regularity at a lower energy scale around 1~TeV. The combined fitting indicates
a Lorentz violation scale ${E}_{\rm LV}=(6.4\pm 1.5)\times10^{17}~{ \rm GeV}$
and an intrinsic time difference ${\Delta {t}_{\rm in}=(-2.8\pm
0.7)\times10^2~{\rm s}}$, from which we find an earlier emission of neutrinos
than photons at the GRB source. We also suggest analyzing neutrino events
detected a few minutes before the GRB trigger time to test the $CPT$ violation
of ultrahigh-energy neutrinos.
