A closed orientable splitting surface in an oriented 3–manifold is a topologically minimal surface of index if its associated disk complex is –connected but not –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.
"An infinite family of links with critical bridge spheres." Algebr. Geom. Topol. 18 (1) 153 - 186, 2018. https://doi.org/10.2140/agt.2018.18.153