  We systematically examine torsional shear oscillations of neutron star crusts
by newly taking into account the possible presence of the phase of cylindrical
nuclei. In this study, we neglect an effect of magnetic fields, under which the
shear oscillations can be damped by the magnetic interaction. First, by
identifying the low frequency quasi-periodic oscillations (QPOs) observed in
the soft-gamma repeaters (SGRs) as the fundamental torsional oscillations, we
constrain the slope parameter of the nuclear symmetry energy, $L$, for
reasonable values of the star's mass $M$ and radius $R$. Meanwhile, we find
that the 1st overtone of torsional oscillations obtained for given $M$ and $R$
can be expressed well as a function of a new parameter $\varsigma\equiv (K_0^4
L^5)^{1/9}$, where $K_0$ is the incompressibility of symmetric nuclear matter.
Assuming that the lowest of the QPO frequencies above 500 Hz observed in SGR
1806-20 comes from the 1st overtone, we can constrain the value of $\varsigma$.
Then, for each neutron star model, such a value of $L$ as can be obtained from
the observed low frequency QPOs translates to the optimal value of $K_0$ via
the above constraint on $\varsigma$. Finally, its consistency with allowed
values of $K_0$ from empirical giant monopole resonances leads to neutron star
models with relatively low mass and large radius, which are qualitatively
similar to the prediction in earlier investigations. This result suggests that
$L\simeq 58-73$ MeV, even when uncertainties in the neutron superfluid density
inside the phase of cylindrical nuclei are allowed for.
