Banded unlink diagrams of surfaces in arbitrary 4-manifolds

Maggie Miller (Princeton)

Abstract: I will describe a system for diagrammatically describing any surface S smoothly embedded in any 4-manifold X via a link with bands attached inside a Kirby diagram for X. These pictures describe the critical points of a Morse function on X restricted to S, analogously to Fox’s movies of surfaces in S^4. We have shown that these diagrams exist and that a simple set of moves (“band moves”) relate any two diagrams of isotopic surfaces. This generalizes work of Swenton and Kearton-Kurlin in S^4 and (with some extra work) confirms a conjecture of Meier and Zupan: that bridge trisections of surfaces in 4-manifolds are unique up to stabilization. I will sketch the proof that two diagrams describing isotopic surfaces are related by band moves (which I will also describe), and possibly define bridge trisections and discuss their uniqueness. This project is joint with Mark Hughes and Seungwon Kim.