---
tag: PW9D9QKAD4
kind: concept
name: Unknotting number
msc: [57K10]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/Unknotting_number
same_as:
  wikidata: Q7896949
  wikipedia: Unknotting number
---

## Summary

The unknotting number of a [[knot|QYEAZTYGPT]] is the least number of crossing changes needed to turn it into the [[unknot|1TQ06R4Q66]].
It is a [[knot invariant|GZQ5ZE3X6G]] that measures how far a knot is from being trivial.
It is easy to define but often hard to compute, and it is still unknown for some knots with only ten crossings.

## Explanations

### L1 human

Think of a knot as a tangled loop of rope, drawn on paper with over- and under-crossings.
A crossing change lets one strand pass through the other at one crossing, so the over-strand becomes the under-strand.
The unknotting number counts how many such passes are needed, at best, to get a plain loop.
You may move the rope around freely between the passes.
The [[trefoil knot|VYTYSVHN4A]] has unknotting number $1$: one crossing change in its usual picture undoes it.