Hopf interlocking

Hopf interlocking
Hopf interlocking

In knot theory , a branch of mathematics , the Hopf link (also Hopf link ) is the simplest example of a link between two circles.

Hopf interlocking

The Hopf link is an entanglement consisting of two unknot (d. H. Unknotted circles), the linking number (depending on the orientation is) plus or minus 1.

A concrete model are, for example, the circles parameterized by and . ${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle (\ cos t, \ sin t, 0)}$${\ displaystyle (\ cos t + 1.0, \ sin t)}$

Topology of complement

The complement of the Hopf link in the 3-sphere is homeomorphic to . The link group , i.e. the fundamental group of the complement, is isomorphic to the free Abelian group with two generators. ${\ displaystyle S ^ {3}}$${\ displaystyle S ^ {1} \ times S ^ {1} \ times \ left (0.1 \ right)}$${\ displaystyle \ mathbb {Z} \ times \ mathbb {Z}}$

Invariants

The Jones polynomial is

${\ displaystyle V (t) = - tt ^ {- 1}}$,

the HOMFLY polynomial is

${\ displaystyle P (z, \ alpha) = z ^ {- 1} (\ alpha ^ {- 1} - \ alpha ^ {- 3}) - z \ alpha ^ {- 1}}$,

the Hopf link is the - torus link and it is the end of the braid . ${\ displaystyle (2.2)}$ ${\ displaystyle \ sigma _ {1} ^ {2}}$

Hopf fiber and homotopy groups

Heinz Hopf examined the Hopf fibers in 1931

${\ displaystyle h \ colon S ^ {3} \ to S ^ {2}}$

and found that every two fibers form a Hopf loop.

In general, he defined the invariant , now known as the Hopf invariant , as the link number of the archetypes of two regular values of and he proved that the assignment ${\ displaystyle f \ colon S ^ {3} \ to S ^ {2}}$${\ displaystyle H (f) \ in \ mathbb {Z}}$${\ displaystyle f}$

Shingon-shu Buzan-ha crest
${\ displaystyle f \ to H (f)}$

an isomorphism

${\ displaystyle \ pi _ {3} (S ^ {2}) \ to \ mathbb {Z}}$

results.

Occurrence in art, science and philosophy

Catenanes
• The Hopf link is from the Shingon shū allocated Buddhist sect Buzan-ha used as an icon.
• Catenanes represent a Hopf link.
• The Hopf link appears in numerous sculptures by the Japanese artist Keizo Ushio .

literature

• Heinz Hopf : About the images of the three-dimensional sphere on the spherical surface. Math. Ann. 104 (1931), 637-665 ( PDF )
• Colin Adams : The Knot Book . Spectrum Academic Publishing House (1995). ISBN 978-3860253380