  We calculate the sensitivity of next generation neutrino telescopes to the 28
(isoscalar and isovector) coupling constants defining the non-relativistic
effective theory of (spin 1/2) dark matter (DM)-nucleon interactions. We take
as a benchmark detector the proposed Precision IceCube Next Generation Upgrade
(PINGU), although our results are valid for any other neutrino telescope of
similar effective volume. We express PINGU's sensitivity in terms of $5\sigma$
sensitivity contours in the DM-mass - coupling constant plane, and compare our
sensitivity contours with the 90% C.L. exclusion limits on the same coupling
constants that we obtain from a reanalysis of the null result of current DM
searches at IceCube/DeepCore. We find that PINGU can effectively probe not only
the canonical spin-independent and spin-dependent DM-nucleon interactions, but
also velocity-dependent or momentum-dependent interactions that generate
coherently enhanced DM-nucleus scattering cross sections. We also find that
PINGU's $5\sigma$ sensitivity contours are significantly below current
IceCube/DeepCore 90% C.L. exclusion limits when $b\bar{b}$ is the leading DM
annihilation channel. This result shows the importance of lowering the
experimental energy threshold when probing models that generate soft neutrino
energy spectra, and holds true independently of the assumed DM-nucleon
interaction and for all DM masses tested here. When DM primarily annihilates
into $\tau\bar{\tau}$, a PINGU-like detector will improve upon current
exclusion limits for DM masses below $35$ GeV, independently of the assumed
DM-nucleon interaction.
