The Jones polynomial for a torus knot with twists

We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form $T((p,q),(2,s))$ where $p$ and $q$ are coprime and $s$ is nonzero. When $s=2n$, these links are the twisted torus knots $T(p,q,2,n)$. We show that for $T(p,q,2,n)$, the Jones polynomial is trivial if and only if the knot is trivial.