  A closed, orientable, splitting surface in an oriented $3$-manifold is a
topologically minimal surface of index $n$ if its associated disk complex is
$(n-2)$-connected but not $(n-1)$-connected. A critical surface is a
topologically minimal surface of index $2$. In this paper, we use an equivalent
combinatorial definition of critical surfaces to construct the first known
critical bridge spheres for nontrivial links.
