  We introduce a new class of links for which we give a lower bound for the
slice genus $g_*$, using the generalized Rasmussen invariant. We show that this
bound, in some cases, allows one to compute $g_*$ exactly; in particular, we
compute $g_*$ for torus links. We also study another link invariant: the strong
slice genus $g_*^*$. Studying the behaviour of a specific type of cobordisms in
Lee homology, a lower bound for $g_*^*$ is also given.
