Hopf link
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2008-05-22T19:58:12Z
ArnoldReinhold
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[[Image:Hopf link.svg|thumb|Hopf link.]]
[[image:skein-relation-link22-plus-sm.png|thumb|right|[[Skein relation]] for the Hopf link.]]
In mathematical [[knot theory]], the '''Hopf link''', named after [[Heinz Hopf]], is the simplest nontrivial [[link (knot theory)|link]] with more than one component. It consists of two [[circle]]s 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 [[orientation]]s of the two components the [[linking number]] of the Hopf link is ±1.
The Hopf link is a (2,2)-[[torus link]] with the [[braid word]]
:<math>\sigma_1^2.\,</math>
In the [[Hopf bundle]]
:<math>S^1 \to S^3 \to S^2.\,</math>
the fibers over any two distinct points in <math>S^2</math> form a Hopf link in the [[3-sphere]] <math>S^3</math>.
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==Gallery==
<gallery>
Image:Moebiusband_wikipedia.png|A Hopf link spanned by an [[annulus]].
Image:Bryce3d-ak109-003.JPG|A [[Computer graphics|computer-generated]] artistic depiction of the Hopf link.
</gallery>
==External links==
*{{MathWorld|urlname=HopfLink|title=Hopf Link}}
*The [http://katlas.math.toronto.edu/wiki/L2a1 Hopf Link] at the wiki [http://katlas.math.toronto.edu/wiki/ Knot Atlas].
{{Knot-stub}}
[[Category:Knot theory]]
[[Category:Eponyms]]
[[fr:Enlacement de Hopf]]