Whitehead Entanglement

The Whitehead link (English: Whitehead link ) is one of the simplest links in the mathematical branch of knot theory .

JHC Whitehead , after whom the Whitehead tangle is named, used it to construct the Whitehead manifold with which he corrected his attempt to prove the Poincaré Conjecture of 1934 himself.

properties

The two components of the Whitehead link have link numbers . ${\ displaystyle 0}$

It is homotopic , but not isotopic to the trivial link. There is an isotopy that reverses the two components of the Whitehead link.

The whitehead loop is the end of the braid ${\ displaystyle \ sigma _ {1} ^ {2} \ sigma _ {2} ^ {2} \ sigma _ {1} ^ {- 1} \ sigma _ {2} ^ {- 2}. \,}$

Your Jones polynomial is ${\ displaystyle V (t) = t ^ {- {3 \ over 2}} (- 1 + t-2t ^ {2} + t ^ {3} -2t ^ {4} + t ^ {5}). }$

Their complement is hyperbolic . A fundamental domain in hyperbolic space is the regular ideal octahedron . The hyperbolic volume of the complement of the Whitehead link is therefore 3.663862377 ..., the volume of the regular ideal octahedron.

The invariant trace body is . ${\ displaystyle \ mathbb {Q} (i)}$

The complements of the Whitehead link and its "sister", the (-2,3,8) - pretzel link , are the two orientable, hyperbolic 3-manifolds of the smallest volume, the edge of which consists of at least two connected components.

By (5,1) - stretching surgery on one of the two components of the Whitehead linkage, the sister manifold of the figure eight complement is obtained , which is one of the two orientable, hyperbolic 3-manifolds of the smallest volume with non-empty margin. Another (5,2) stretching surgery on the remaining component gives the weeks manifold , which is the (closed) hyperbolic 3-manifold of the smallest volume.

Web links

Whitehead Link in the Knot Atlas
Whitehead Link in MathWorld

Individual evidence

↑ Whitehead, A certain open manifold whose group is unity , Quarterly journal of mathematics 6, 1935, pp. 268-279
^ Whitehead, Certain theorems about three-dimensional manifolds (I) , Quarterly journal of mathematics, Volume 5, 1934, pp. 308-320
↑ William Thurston : The Geometry and Topology of Three-Manifolds , Chapter 3.3, online (pdf)
^ Ian Agol : The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3723-3732.

[1] Whitehead, A certain open manifold whose group is unity , Quarterly journal of mathematics 6, 1935, pp. 268-279

[2] Whitehead, Certain theorems about three-dimensional manifolds (I) , Quarterly journal of mathematics, Volume 5, 1934, pp. 308-320

[3] William Thurston : The Geometry and Topology of Three-Manifolds , Chapter 3.3, online (pdf)

[4] Ian Agol : The minimal volume orientable hyperbolic 2-cusped 3-manifolds. Proc. Amer. Math. Soc. 138 (2010), no. 10, 3723-3732.