nLab knot complement

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)

Related concepts

• hyperbolic knot

References

Exposition:

• Abhijit Champanerkar, The geometry of knot complements (pdf, pdf)

Lecture notes:

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

• Zach Haney, Knot Complements, 2016 (pdf)

See also

• Wikipedia, Knot complement

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

• Carl-Friedrich Bödigheimer, Fred Cohen, L. Taylor, Examples 5.2 of: On the homology of configuration spaces, Topology Vol. 28 No. 1, p. 111-123 1989 (pdf)