---
tag: 727ZEPVSTC
kind: concept
name: Kontsevich integral
msc: [57K16]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/Kontsevich_invariant
same_as:
  wikidata: Q623129
  wikipedia: Kontsevich invariant
---

## Summary

The Kontsevich integral is a [[knot invariant|GZQ5ZE3X6G]] with values in a completed algebra of chord diagrams.
Maxim Kontsevich defined it by an iterated integral of logarithmic forms.
It is the universal rational [[finite type invariant|TWH3WVKFF1]]: every such invariant is obtained from it by a linear map.
It ties knot theory to quantum groups and to the theory of Drinfeld associators.

## Explanations

### L1 human

Place a knot in space and slice it by horizontal planes.
Each slice meets the knot in finitely many points, which move as the plane goes up.
The Kontsevich integral records how these points wind around each other, using chords that join pairs of them.
The output is not a number but a formal sum of chord diagrams, circles with chords drawn inside.
From this one object, every rational finite type invariant can be read off.