---
tag: W890NRHGS4
kind: concept
name: Dehn twist
msc: [57K20, 57R50]
sources:
  - url: https://en.wikipedia.org/wiki/Dehn_twist
same_as:
  wikidata: Q5252320
  wikipedia: Dehn twist
---

## Summary

A Dehn twist is a self-homeomorphism of a [[surface|VGJJVBZ1MQ]].
It cuts the surface along a simple closed curve, twists one side by a full turn, and glues back.
It is the identity outside a thin annulus around the curve.
Dehn twists are the basic elements of the [[mapping class group|JV5NYG913V]] of a surface: by the [[Dehn–Lickorish theorem|BG3EH18KRE]] finitely many of them generate it.

## Explanations

### L1 human

Take a garden hose, hold one end fixed, and turn the other end once around.
The hose itself is twisted in a narrow band, and away from that band nothing moves.
On a surface, do the same in a thin band around a loop.
A straight line crossing the band gets wrapped once around the loop.
On a torus, a twist around the meridian turns the longitude into a curve that goes once around each way.