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...

In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)...

nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles (or unknots) linked together once. The circles...

gives the unknot and the Hopf link. Applying the relation to the Hopf link where indicated, C() = C() + z C() gives a link deformable to one with 0 crossings...

Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of dynamical systems, topology and geometry. Hopf was born...

crossings (the Hopf link [1]). Patching the trefoil P() = A × P() + P() gives the unknot and, again, the Hopf link. Patching the Hopf link P() = A × P()...

Hopf is a German surname. Notable people with the surname include: Eberhard Hopf (1902–1983), Austrian mathematician Hans Hopf (1916–1993), German tenor...

In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic...

mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important...

trefoil knot, and figure-eight knot are fibered knots. The Hopf link is a fibered link. The Alexander polynomial of a fibered knot is monic, i.e. the...

This link cobordism between the Hopf link and the unlink is topologically a pair of pants....

Differentiable curve – Study of curves from a differential point of view Hopf invariant – Homotopy invariant of maps between n-spheres Kissing number –...

unknot. The link group of an n-component unlink is the free group on n generators, and is used in classifying Brunnian links. The Hopf link is a simple...

integers, and the link group of an unlinked union is the free product of the link groups of the components. The link group of the Hopf link, the simplest...

whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants...

and over each other. This contrasts with two crossings in the simpler Hopf link. In most artistic representations, the parts of the loops that alternately...

the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map η...

each other, nest into each other and pass through each other forming a Hopf link. The 16 triangle faces can be seen in a 2D net within a triangular tiling...

are also linked, not in a Möbius loop but as a Hopf link of two non-intersecting circles, as are all the Clifford parallel isoclines of a Hopf fiber bundle...

Karl Hopf may refer to: Karl Hopf (historian) Karl Hopf (serial killer) This disambiguation page lists articles about people with the same name. If an...

theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings, from...

no such continuous motion, the two curves are said to be linked. For example, the Hopf link is formed by two circles that each pass through the disk spanned...

are also linked, not in a Möbius loop but as a Hopf link of two non-intersecting circles, as are all the Clifford parallel isoclines of a Hopf fiber bundle...

knot theory. The periodic orbits with winding numbers 1 and 2 form a Hopf link, showing that no diffeomorphism can separate these orbits. The banding...

knot K ′ {\displaystyle K'} with a Hopf link. If the link K ′ ∪ J {\displaystyle K'\cup J} is the Whitehead link, K {\displaystyle K} is called a Whitehead...

simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted...

In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be...

1 link/Unlink - equivalent under ambient isotopy to finitely many disjoint circles in the plane 22 1 link/Hopf link - the simplest nontrivial link with...

is needed to determine the usage. There are analogous definitions for the link complement. Many knot invariants, such as the knot group, are really invariants...

simplest Brunnian link Brunnian link, a set of links which become trivial if one loop is removed Hopf link, the simplest non-trivial link Solomon's knot...