  We present the first searches for gravitational waves from r-modes of the
Crab pulsar, coherently and separately integrating data from three stretches of
the first two observing runs of Advanced LIGO using the F-statistic. The second
run was divided in two by a glitch of the pulsar roughly halfway through. The
frequencies and derivatives searched were based on radio measurements of the
pulsar's spin-down parameters as described in Caride et al., Phys. Rev. D 100,
064013 (2019). We did not find any evidence of gravitational waves. Our best
90% confidence upper limits on gravitational wave intrinsic strain were 1.5e-25
for the first run, 1.3e-25 for the first stretch of the second run, and 1.1e-25
for the second stretch of the second run. These are the first upper limits on
gravitational waves from r-modes of a known pulsar to beat its spin-down limit,
and they do so by more than an order of magnitude in amplitude or two orders of
magnitude in luminosity.
