---
tag: SWJZ4QRF9W
kind: concept
name: Crossing number
msc: [57K10]
sources:
  - url: https://en.wikipedia.org/wiki/Crossing_number_(knot_theory)
same_as:
  wikidata: Q2548661
  wikipedia: Crossing number (knot theory)
---

## Summary

The crossing number of a [[knot|QYEAZTYGPT]] is the least number of crossings in any diagram of the knot.
It is a [[knot invariant|GZQ5ZE3X6G]] with values in the natural numbers, and knot tables list knots by their crossing number.
It is easy to define but hard to compute, because one must rule out all diagrams with fewer crossings.

## Explanations

### L1 human

Draw a knot on paper, with a gap wherever one strand passes under another.
Different drawings of the same knot can have different numbers of crossings.
The crossing number is the smallest number that any drawing can achieve.
The unknot has crossing number $0$, the [[trefoil|VYTYSVHN4A]] has $3$, and the [[figure-eight knot|C01XBGAHC6]] has $4$.