  Long predicted more than fifty years ago, strong evidence for the existence
of crystalline cores inside white dwarfs has recently been obtained by the Gaia
space telescope. It is thus important to investigate how a crystalline core may
affect the properties and dynamics of white dwarfs. In this paper, we first
study the dependence of the frequencies of the fundamental (f), interfacial
(i), and shear (s) oscillation modes on the size of the crystalline core. We
find that the frequencies of the i- and s-modes depend sensitively on the size
of the core, while the frequency of the f-mode is affected only slightly by at
most a few percent for our chosen white dwarf models. We next consider the
tidal deformability of crystallized white dwarfs and find that the effect of
crystallization becomes significant only when the radius of the core is larger
than about 70% of the stellar radius. The tidal deformability can change by a
few to about 10 percent when a white dwarf becomes fully crystallized. We also
show that there exist approximate equation-of-state insensitive relations
connecting the mass, moment of inertia, tidal deformability, and f-mode
frequency for pure fluid white dwarfs. Depending on the stellar mass and
composition, however, these relations can be affected by a few percent when the
white dwarf is crystallized. These changes could leave an imprint on the
gravitational waves emitted from the late inspiral or merger of white dwarf
binaries, which may be detectable by future space-borne gravitational wave
detectors.
