Symplectic surfaces and bridge position

Abstract

We give a new characterization of symplectic surfaces in \({\mathbb{C}\mathbb{P}}^2\) via bridge trisections. Specifically, a minimal genus surface in \({\mathbb{C}\mathbb{P}}^2\) is smoothly isotopic to a symplectic surface if and only if it is smoothly isotopic to a surface in transverse bridge position. We discuss several potential applications, including the classification of unit 2-knots, establishing the triviality of Gluck twists, the symplectic isotopy problem, Auroux’s proof that every symplectic 4-manifold is a branched cover over \({\mathbb{C}\mathbb{P}}^2\), and the existence of Weinstein trisections. The proof exploits a well-known connection between symplectic surfaces and quasipositive factorizations of the full twist in the braid group.