---
tag: SFJ329YB9F
kind: concept
name: Linking number
msc: [57K10, 55M25]
sources:
  - url: https://en.wikipedia.org/wiki/Linking_number
same_as:
  wikidata: Q2000614
  wikipedia: Linking number
---

## Summary

The linking number of two disjoint oriented closed curves in $\R^3$ is an integer that counts how many times one curve winds around the other.
It does not change when the curves move without crossing each other, so it is an invariant of two-component [[links|1A5WRCNTM3]].
It can be read off a diagram, computed by a homology class, or given by the Gauss double integral.

## Explanations

### L1 human

Take two closed loops of string in space that do not touch.
The linking number says how many times the first loop goes through the second, counted with a sign for the direction.
Two separate rings that can be pulled apart have linking number $0$.
Two rings hooked once, as in a chain, have linking number $1$ or $-1$, depending on the chosen directions.
You cannot unhook the rings without cutting one of them, and the number shows this.