Hopf link

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Skein relation for the Hopf link.
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Skein relation for the Hopf link.

In mathematical knot theory, the Hopf link, named after Heinz Hopf, is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once. For a concrete model take the unit circle in the xy-plane centered at the origin and another unit circle in the yz-plane centered at (0,1,0).

Depending on the relative orientations of the two components the linking number of the Hopf link is ±1.

A Hopf link spanned by an annulus (image created with povray).
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A Hopf link spanned by an annulus (image created with povray).

The Hopf link is a (2,2)-torus link with the braid word

\sigma_1^2.\,

In the Hopf bundle

S^1 \to S^3 \to S^2.\,

the fibers over any two distinct points in S2 form a Hopf link in the 3-sphere S3.


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