Hopf link

In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. In mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. A concrete model consists of two unit circles in perpendicular planes, each passing through the center of the other. This model minimizes the ropelength of the link and until 2002 the Hopf link was the only link whose ropelength was known. The convex hull of these two circles forms a shape called an oloid. Depending on the relative orientations 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 The knot complement of the Hopf link is R × S1 × S1, the cylinder over a torus. This space has a locally Euclidean geometry, so the Hopf link is not a hyperbolic link. The knot group of the Hopf link (the fundamental group of its complement) is Z2 (the free abelian group on two generators), distinguishing it from an unlinked pair of loops which has the free group on two generators as its group.