nLab
knot complement

Contents

Contents

Idea

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

Properties

Homology

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

H n(S 3K,)={ | n{0,1} 0 | n2 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)

See also

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

Last revised on October 7, 2019 at 01:51:33. See the history of this page for a list of all contributions to it.