Lecture notes on link homologies and knotted surfaces

TL;DR

This paper provides an overview of link homology theories like Khovanov and knot Floer homology, focusing on their applications to knotted surfaces in 4-space and computational techniques for these invariants.

Contribution

It offers a survey of the formal properties and applications of link homologies to knotted surfaces, including computational methods and the role of the Bar-Natan category.

Findings

Survey of link homology theories and their formal properties

Introduction to cobordism maps in Khovanov homology

Hands-on computational techniques for Khovanov and Bar-Natan homology

Abstract

Link homology theories (such as knot Floer homology and Khovanov homology) have become indispensable tools for studying knots and links, including powerful 4-dimensional obstructions. These notes, based on lectures given at the 2024 Georgia Topology Summer School, discuss what these toolkits say about surfaces in 4-space themselves, via the homomorphisms assigned to link cobordisms. We begin with a brief overview of these theories (focusing on their shared formal properties) and survey some of their applications to knotted surfaces. Afterwards, we give an introduction to Khovanov homology (with an eye towards its cobordism maps), discuss hands-on computational techniques for Khovanov and Bar-Natan homology, and outline the role of the Bar-Natan category in this story.