---
tag: XB24W24B20
kind: concept
name: Khovanov homology
msc: [57K18]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/Khovanov_homology
same_as:
  wikidata: Q6402652
  wikipedia: Khovanov homology
---

## Summary

Khovanov homology is an invariant of oriented [[links|1A5WRCNTM3]] that takes values in bigraded abelian groups.
Mikhail Khovanov built it from a cube of smoothings of a diagram, as a categorification of the [[Jones polynomial|4SPC17FPXE]].
Its graded Euler characteristic is the Jones polynomial, but it carries strictly more information.
Kronheimer and Mrowka proved that it detects the [[unknot|1TQ06R4Q66]].

## Explanations

### L1 human

The [[Kauffman bracket|JYKVYS4N32]] computes the Jones polynomial by summing over all ways to smooth the crossings of a diagram.
Khovanov homology replaces each term of that sum by a group and each sign by a map between groups.
The result is a chain complex, and its homology does not depend on the diagram.
Counting dimensions with signs gives back the Jones polynomial.
The extra structure lets it tell apart knots that the Jones polynomial cannot.