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.