---
tag: 5P608WRETM
kind: concept
name: Knot Floer homology
msc: [57K18, 57R58]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/Floer_homology
---

## Summary

Knot Floer homology is an invariant of [[knots|QYEAZTYGPT]] in $3$-manifolds built from [[Heegaard Floer homology|2WJJBN2T5P]].
It was introduced by Ozsváth and Szabó, and independently by Rasmussen.
For a knot in $S^3$ it is a bigraded vector space whose graded Euler characteristic is the [[Alexander polynomial|JKA25KDPQP]].
It detects the Seifert genus of a knot and whether the knot is fibered.

## Explanations

### L1 human

Knot Floer homology describes a knot by a finite list of vector spaces, arranged in a grid by two gradings.
Adding up their dimensions with suitable signs recovers the Alexander polynomial.
But the list knows more: the highest row that is not zero tells you the least genus of a surface bounded by the knot.
It comes from counting certain holomorphic disks, a technique from symplectic geometry.