  Transverse oscillations of small amplitude are commonly seen in
high-resolution observations of filament threads, i.e. the fine-structures of
solar filaments/prominences, and are typically damped in a few periods. Kink
wave modes supported by the thread body offer a consistent explanation of these
observed oscillations. Among the proposed mechanisms to explain the kink mode
damping, resonant absorption in the Alfven continuum seems to be the most
efficient as it produces damping times of about 3 periods. However, for a
nonzero-beta plasma and typical prominence conditions, the kink mode is also
resonantly coupled to slow (or cusp) continuum modes, which could further
reduce the damping time. In this Letter, we explore for the first time both
analytically and numerically the effect of the slow continuum on the damping of
transverse thread oscillations. The thread model is composed of a homogeneous
and straight cylindrical plasma, an inhomogeneous transitional layer, and the
homogeneous coronal plasma. We find that the damping of the kink mode due to
the slow resonance is much less efficient than that due to the Alfven
resonance.
