A trisection (introduced by Gay and Kirby) is a certain decomposition of a smooth, closed, orientable 4-manifold $X^4$ which can be described by sets of curves on a surface. Meier and Zupan showed that a trisection of $X^4$ can be used to encode any surface smoothly embedded in $X^4$, in a manner analogous to a bridge splitting of a knot. They showed that these bridge trisections are unique for surfaces in $S^4$ (with respect to a standard trisection), up to perturbation. I will sketch why bridge trisections are unique up to perturbation even in the general case, after giving some basic definitions and exposition.

This is joint work with Mark Hughes and Seungwon Kim.