An introduction to knot Floer homology

TL;DR

This survey introduces knot Floer homology, detailing its multiple constructions, geometric insights, and connections to other topological invariants like Khovanov-Rozansky homology.

Contribution

It provides a comprehensive overview of knot Floer homology, including new combinatorial descriptions and insights into its topological significance.

Findings

Multiple constructions of knot Floer homology presented

Connections to 3- and 4-dimensional topology discussed

Potential relations to Khovanov-Rozansky homology explored

Abstract

This is a survey article about knot Floer homology. We present three constructions of this invariant: the original one using holomorphic disks, a combinatorial description using grid diagrams, and a combinatorial description in terms of the cube of resolutions. We discuss the geometric information carried by knot Floer homology, and the connection to three- and four-dimensional topology via surgery formulas. We also describe some conjectural relations to Khovanov-Rozansky homology.