A knot-theoretic tour of dimension four

TL;DR

This paper provides an overview of four-dimensional manifold topology, linking knot theory with contact, symplectic, and complex geometry, aimed at graduate students in singularity theory and low-dimensional topology.

Contribution

It offers a comprehensive introduction connecting knot theory, contact, symplectic, and complex geometry within four-dimensional topology, highlighting recent developments and pedagogical insights.

Findings

Connections between knot theory and 4-manifold topology

Insights into Stein surfaces and complex geometry

Educational overview for graduate students

Abstract

These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the R\'enyi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.