To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

# Whitehead link

## From Wikipedia, the free encyclopedia

Simple depiction
Old Thor's hammer archaeological artefact

In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links.

Whitehead spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, the Whitehead link was used as part of his construction of the now-named Whitehead manifold, which refuted his previous purported proof of the conjecture.

## Structure

The link is created with two projections of the unknot: one circular loop and one figure eight-shaped loop (i.e., a loop with a Reidemeister Type I move applied) intertwined such that they are inseparable and neither loses its form. Excluding the instance where the figure eight thread intersects itself, the Whitehead link has four crossings. Because each underhand crossing has a paired upperhand crossing, its linking number is 0. It is not isotopic to the unlink, but it is link homotopic to the unlink.

In braid theory notation, the link is written

${\displaystyle \sigma _{1}^{2}\sigma _{2}^{2}\sigma _{1}^{-1}\sigma _{2}^{-2}.\,}$

Its Jones polynomial is

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

This polynomial and ${\displaystyle V(1/t)}$ are the two factors of the Jones polynomial of the L10a140 link. Notably, ${\displaystyle V(1/t)}$ is the Jones polynomial for the mirror image of a link having Jones polynomial ${\displaystyle V(t)}$.

## Volume

The hyperbolic volume of the complement of the Whitehead link is 4 times Catalan's constant, approximately 3.66. The Whitehead link complement is one of two two-cusped hyperbolic manifolds with the minimum possible volume, the other being the complement of the pretzel link with parameters (−2,3,8).[1]

Dehn filling on one component of the Whitehead link can produce the sibling manifold of the complement of the figure-eight knot, and Dehn filling on both components can produce the Weeks manifold, respectively one of the minimum-volume hyperbolic manifolds with one cusp and the minimum-volume hyperbolic manifold with no cusps.

## References

1. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:, doi:10.1090/S0002-9939-10-10364-5, MR 2661571.
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.