  We reveal an intimate connection between the torus link $T_{2s,2t}$ and the
logarithmic $(s,t)$ VOA. We show that the singlet character of $(s,t)$-log VOA
at the root of unity coincides with the Kashaev invariant and that it has a
property of the quantum modularity. Also shown is that the tail of the
$N$-colored Jones polynomial gives the character. Furthermore we propose a
geometric method to compute the character.
