From knots to four-manifolds

TL;DR

This survey explores the relationship between knot theory and four-manifold topology, highlighting how link representations and invariants have advanced understanding of smooth four-manifolds and exotic structures.

Contribution

It reviews progress in using Kirby diagrams, Heegaard Floer theory, and skein modules to analyze four-manifolds and proposes a program to study four-manifolds via boundary knots.

Findings

Progress in computing smooth four-manifold invariants

Development of skein lasagna modules for four-manifold analysis

Potential to understand four-manifolds through boundary knot properties

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.