## Diagrams for surface isotopies in $$4$$-manifolds

Maggie Miller (Princeton)

Abstract

I will demonstrate a method of drawing diagrams for a surface smoothly embedded into an arbitrary $$4$$-manifold, and show that any two diagrams of smoothly isotopic surfaces are related by a sequence of simple moves (6 elementary moves + isotopy). This generalizes work in $$S^4$$ of Swenton and Kearton-Kurlin. Through correspondence with bridge trisections, this implies that a surface in a trisected $$4$$-manifold has a unique bridge trisection up to perturbation, proving a conjecture of Meier and Zupan.