  Deconfined quark matter may exist in a crystalline color-superconducting
phase in the interiors of compact stars. In this paper, we study the torsional
oscillations of compact stars featuring a crystalline color-superconducting
quark-matter core in general relativity. Depending on the size of the
crystalline core and the value of the gap parameter $\Delta$, we find that the
frequencies of the torsional oscillation modes can range from a few hundred
hertz to a few kilohertz for our canonical $1.4 M_\odot$ compact star models.
We have also studied the prospect for detecting the gravitational-wave signals
emitted from these modes in a pulsar glitch event. Assuming that at least
$10\%$ of the energy released in a Vela glitch can be channeled to the
oscillation modes, we find that the Einstein Telescope should be able to detect
these signals in quite general situations. Furthermore, if the size of the
crystalline core is comparable to the stellar radius and the gap parameter is
relatively small at $\Delta \sim 5$ MeV, the signal-to-noise ratio for Advanced
LIGO could reach $\sim$10 for a Vela glitch. Our optimistic results suggest
that we might already be able to probe the nature of crystalline
color-superconducting quark matter with the second-generation
gravitational-wave detectors when they come online in the next few years.
