---
tag: NS60VV786W
kind: concept
name: Torus knot
msc: [57K10]
specializes: [QYEAZTYGPT]
sources:
  - url: https://en.wikipedia.org/wiki/Torus_knot
same_as:
  wikidata: Q1892897
  wikipedia: Torus knot
---

## Summary

A torus knot is a [[knot|QYEAZTYGPT]] that lies on the surface of a standard unknotted [[torus|03NM7DZXXC]] in space.
For coprime integers $p$ and $q$, the torus knot $T(p, q)$ winds $p$ times around the axis of the torus and $q$ times around its tube.
Torus knots form one of the three classes in Thurston's division of knots into torus, [[satellite|A1E8TDEVJ9]] and [[hyperbolic|K47G2Q24DT]] knots.
Many invariants of torus knots can be computed exactly.

## Explanations

### L1 human

Picture a doughnut and wrap a string around its surface without crossing itself.
Go around the hole $p$ times while going through the hole $q$ times, then close the string up.
When $p$ and $q$ have no common factor you get a single loop, the torus knot $T(p, q)$.
The [[trefoil knot|VYTYSVHN4A]] is $T(2, 3)$: twice around the hole and three times through it.
If $p$ or $q$ is $\pm 1$, the loop can be slid off the doughnut and untied.