An introduction to knot Floer homology

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 dia- grams, and a combinatorial description in terms of the cube of resolutions. We discuss the geomet- ric 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. Knot Floer homology is an invariant of knots and links in three-manifolds. It was introduced independently by Ozsvath-Szabo (OS04c) and Rasmussen (Ras03) around 2002. Since then it has grown into a large subject. Its importance lies in the fact that it contains information about several non-trivial geometric properties of the knot (genus, slice genus, fiberedness, effects of surgery, etc.) Furthermore, knot Floer homology is computable: There exist general algorithms that can calculate it for arbitrary knots. These algorithms tend to get slow as the complexity of the knot increases, but there are also different methods that can be applied to special classes of knots and give explicit