Hopf link 1515511 214260276 2008-05-22T19:58:12Z ArnoldReinhold 84951 Show simple image first, add Gallery [[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 &plusmn;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>. <br style="clear:both;"> ==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]]