Hopf link

In knot theory, a branch of mathematics, the Hopf - interlacing (also Hopf link), the simplest example of an entanglement of two circles.

Hopf entanglement

The Hopf entanglement is an entanglement of two unknot (ie unknotted circles ) whose linking number is (depending on orientation) plus or minus 1.

A concrete model, for example in the by and parameterized loops.

Topology of the complement

The complement of the Hopf entanglement in the 3- sphere is homeomorphic. The link group, ie the fundamental group of the complement is isomorphic to the free abelian group with two generators.

Invariants

The Jones polynomial is

HOMFLY the polynomial is

The Hopf entanglement is the torus link and it is the conclusion of the braid.

Hopf fibration and homotopy groups

Heinz Hopf in 1931 examined the Hopf fibration

And noted that any two fibers form a Hopf entanglement.

Generally, it is defined for the pictures today known as the Hopf invariant invariant as a linking number of the preimages of two regular values ​​of and he proved that the mapping

An isomorphism

Results.

Occurrences in art, science and philosophy

  • The Hopf entanglement is of the allocable to the Shingon Buddhist sect shū Buzan -ha used as a symbol.
  • Catenanes represent a Hopf entanglement
  • The Hopf entanglement occurs in numerous sculptures of Japanese artist Keizo Ushio.
Link (knot theory) Knot group Torus knot Braid group Buddhism
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