From knots to four-manifolds

Abstract

This is a survey article about the connections between knot theory and four-dimensional topology. Every four-manifold can be represented in terms of a link, by a Kirby diagram. This point of view has led to progress in computing invariants of smooth four-manifolds that can detect exotic structures. We explain how this was done in two contexts: Heegaard Floer theory and skein lasagna modules. We also describe a program to understand four-manifolds through the properties of knots on their boundaries.

29 pages, 19 figures; to appear in Proceedings of the ICM 2026