---
tag: Q468P6FP4A
kind: definition
name: Dehn surgery
msc: [57K30, 57K10]
defines: ZA405B0Z1S
uses: [R20TB2KEQQ, QYEAZTYGPT, 1A5WRCNTM3, 0QT00PB5G2]
sources:
  - url: https://en.wikipedia.org/wiki/Dehn_surgery
---

## Definition

Let $M$ be a closed orientable [[3-manifold|R20TB2KEQQ]] and let $K \subset M$ be a smooth [[knot|QYEAZTYGPT]], that is, a smoothly embedded circle.
Let $N(K)$ be a closed tubular neighbourhood of $K$, a solid torus, and let $E = M \setminus \operatorname{int} N(K)$ be the knot exterior, with boundary torus $\partial E = \partial N(K)$.
Let $\alpha \subset \partial E$ be an essential simple closed curve, called the *slope*.
Let $V = D^2 \times S^1$ and let $h \colon \partial V \to \partial E$ be a [[homeomorphism|0QT00PB5G2]] with $h(\partial D^2 \times \set{1}) = \alpha$.
The *Dehn surgery* on $K$ with slope $\alpha$ is the manifold
$$M_K(\alpha) = E \cup_h V.$$
Its homeomorphism type depends only on the isotopy class of $\alpha$ in $\partial E$, not on its orientation.
For a [[link|1A5WRCNTM3]] one does this on each component, with one slope for each.

## Remarks

For a knot $K$ in $S^3$, let $\mu$ be a meridian, bounding a disc in $N(K)$, and $\lambda$ a longitude that is null-homologous in $E$, both oriented.
Each slope is isotopic to a curve representing $p\mu + q\lambda$ in $H_1(\partial E; \Z)$ with $p, q$ coprime, and the slope is labelled by $p/q \in \Q \cup \set{\infty}$.
The slope $\infty = 1/0$ gives back $S^3$; a slope with $q = \pm 1$ is *integral*.

Example: let $U$ be the unknot in $S^3$.
Its exterior is a solid torus in which $\lambda$ bounds a disc.
Surgery with slope $0$, that is $\alpha = \lambda$, glues two solid tori so that their meridians match, and the result is $S^2 \times S^1$.
Surgery on $U$ with slope $p/q$, $p \ne 0$, gives a lens space $L(p, q)$, up to the orientation convention.

Non-example: a curve $\alpha$ that bounds a disc in $\partial E$ is not a slope.
No homeomorphism $h$ can send $\partial D^2 \times \set{1}$ to it, because a homeomorphism of tori sends essential curves to essential curves.

## Explanations

### L1 human

Drill out a thin tube around a knot and fill the hole with a solid tube glued in a new way.
The only choice is which loop on the boundary torus becomes the boundary of a disc of the new tube.
This choice is a fraction, and different fractions usually give different spaces.