  In this paper, we study surfaces embedded in $4$-manifolds. We give a
complete set of moves relating banded unlink diagrams of isotopic surfaces in
an arbitrary $4$-manifold. This extends work of Swenton and Kearton-Kurlin in
$S^4$. As an application, we show that bridge trisections of isotopic surfaces
in a trisected $4$-manifold are related by a sequence of perturbations and
deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also
exhibit several isotopies of unit surfaces in $\mathbb{C}P^2$ (i.e. spheres in
the generating homology class), proving that many explicit unit surfaces are
isotopic to the standard $\mathbb{C}P^1$. This strengthens some previously
known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.

Geometric Topology