Manifolds and cell complexes Knot theory Example

Whitehead link

Example

Start with a round circle $C_1$ in $\R^3$ , and let $D$ be the flat disk that it bounds. Take a second closed curve $C_2$ , disjoint from $C_1$ , that goes down through $D$ and then comes back up through $D$ , with one twist. In the standard diagram, $C_2$ has the shape of a figure eight with one self-crossing, and it clasps $C_1$ . The union $C_1 \cup C_2$ is a link with $2$ components, the Whitehead link. Its standard diagram has $5$ crossings: one crossing of $C_2$ with itself, and four crossings between $C_1$ and $C_2$ . The four crossings between the components come in two pairs of opposite signs, so the linking number of $C_1$ and $C_2$ is $0$ . Each component on its own is an unknot.

Remarks

Although its linking number is $0$ , the Whitehead link is not the unlink of two components: the two curves cannot be separated. One way to see this uses its complement in $S^3$ , which carries a complete hyperbolic metric of finite volume. The complement of a split link contains a sphere that separates the components. A hyperbolic complement contains no such sphere. The complement is built from one regular ideal octahedron, and its volume is about $3.6639$ . Its crossing number is $5$ . It was studied by J. H. C. Whitehead in his work on contractible open $3$ -manifolds. Together with the Hopf link, it shows that the linking number detects some but not all linking.

Tag	`TPGBASPKE0`
URI	`m2://TPGBASPKE0`
Kind	example
Writing	W0 stub
MSC	57K10, 57K32
Instance of	Link
Same as	Wikidata Q2588098 Wikipedia Whitehead link
Changed	2 November 2026

Whitehead link

Example

Remarks

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