---
tag: AVQ9WN1MDV
kind: concept
name: Kirby calculus
msc: [57K40, 57K30, 57R65]
sources:
  - url: https://en.wikipedia.org/wiki/Kirby_calculus
same_as:
  wikidata: Q5956316
  wikipedia: Kirby calculus
---

## Summary

Kirby calculus is a set of moves on framed link diagrams in the $3$-sphere that relate any two diagrams describing the same $3$-manifold.
A framed [[link|1A5WRCNTM3]] describes a $4$-manifold, obtained by attaching $2$-handles to the $4$-ball along the link, and its boundary $3$-manifold, obtained by [[Dehn surgery|ZA405B0Z1S]].
Robion Kirby proved that two framed links give the same oriented $3$-manifold if and only if they are related by his moves.
It is a basic working tool of $3$- and $4$-manifold topology, used to construct manifolds and to prove that two of them are diffeomorphic.

## Explanations

### L1 human

Every closed orientable $3$-dimensional space can be drawn as a picture of knotted loops in ordinary space, each labelled by an integer.
Many different pictures describe the same space.
Kirby calculus is the list of allowed picture changes, a little like the rules for simplifying an algebraic expression.
For example, a single unknotted loop labelled $+1$ or $-1$, far away from the other loops, can be added or erased.
This does not change the space, and the loop alone describes the $3$-sphere.