# nLab knot complement

Contents

### Context

#### Knot theory

knot theory

Examples/classes:

knot invariants

Related concepts:

category: knot theory

# Contents

## Idea

By a knot complement one means the complement of a knot. More precisely: The complement in the 3-sphere $S^{3}$ or in Euclidean space/Cartesian space $\mathbb{R}^{3}$ of a tubular neighbourhood of a knot.

## Properties

### Homology

The ordinary homology of the complement of a knot $K \colon S^1 \hookrightarrow S^3$ in the 3-sphere is concentrated in degrees 0 and 1:

$H_n\big( S^3\setminus K, \mathbb{Z}\big) \;=\; \left\{ \array{ \mathbb{Z} &\vert& n \in \{0,1\} \\ 0 &\vert& n \geq 2 } \right.$

(see for instance Greene 13, Prop. 3.1, Haney 16, Cor. 1.6)

## References

Exposition:

Lecture notes:

• Joshua Greene, Combinatorial Methods in Knot Theory, 2013 (pdf)

• Zach Haney, Knot Complements, 2016 (pdf)

On the ordinary homology of configuration spaces of points in knot complements in $\mathbb{R}^3$: