In 2015, Jeff Meier and I introduced bridge trisections of knotted surfaces in the 4-sphere, a natural adaptation of Gay-Kirby trisections. Bridge trisections also give rise to tri-plane diagrams, a new way of doing knotted surface theory that reflects classical knot theory. In later work, we extended the definition of bridge trisections to surfaces in arbitrary 4-manifolds. We will survey some examples, current results, and open questions.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/TopologySeminar