---
tag: ZA405B0Z1S
kind: concept
name: Dehn surgery
msc: [57K30, 57K10]
sources:
  - url: https://en.wikipedia.org/wiki/Dehn_surgery
same_as:
  wikidata: Q3675173
  wikipedia: Dehn surgery
---

## Summary

Dehn surgery builds a new [[3-manifold|R20TB2KEQQ]] from a [[knot|QYEAZTYGPT]] or [[link|1A5WRCNTM3]] in a given one.
One removes a solid torus around the knot.
Then one glues it back so that its meridian disc caps off a chosen curve on the boundary torus.
For knots in $S^3$ the choices are labelled by a number $p/q \in \Q \cup \set{\infty}$.
By the [[Lickorish–Wallace theorem|1YP4RF1QXV]], every closed orientable $3$-manifold arises by Dehn surgery on a link in $S^3$.

## Explanations

### L1 human

Think of a knotted rope inside space, wrapped in a thin solid tube.
Remove the tube and put a solid tube back in, but twisted.
Now the disc that used to go around the rope closes off a different loop on the tube's surface.
The space you get may be completely new.
For the simplest knot, the round circle, surgery with label $0$ gives the product of a sphere and a circle.