Date: Tuesday April 16, 2019

Speaker: Maggie Miller (Princeton)

Title: Banded unlink diagrams of surfaces

Abstract: The study of surfaces smoothly embedded into 4-manifolds is a natural higher-dimensional analogue of knot theory. Surface embeddings can encode information about the smooth structure or various complexities of a 4-manifold, but we are constrained by our ability to describe such embeddings. In this talk, I will describe how to visualize and diagram a surface in $S^4$ via a "banded unlink diagram." This is essentially a diagrammatic depiction of how a standard Morse function on $S^4$ restricts to a given surface. Swenton and later Kearton and Kurlin showed that there is a set of simple moves on these banded unlink diagrams that relate any two diagrams describing isotopic surfaces. I will sketch this proof.

Recently Mark Hughes and Seungwon Kim and I have generalized this proof of uniqueness to diagrams of surfaces in arbitrary 4-manifolds (with a few extra types of move necessary). I will discuss the statement and proof of this theorem in the topology seminar.