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
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

In knot 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 the overlay of a circle and a figure-eight shaped loop.

Structure

Alternating link diagram
Alternative diagram, symmetric by 3d rotation around a vertical line in the plane of the drawing[1]

A common way of describing this knot is formed by overlaying a figure-eight shaped loop with another circular loop surrounding the crossing of the figure-eight. The above-below relation between these two unknots is then set as an alternating link, with the consecutive crossings on each loop alternating between under and over. This drawing has five crossings, one of which is the self-crossing of the figure-eight curve, which does not count towards the linking number. Because the remaining crossings have equal numbers of under and over crossings on each loop, its linking number is 0. It is not isotopic to the unlink, but it is link homotopic to the unlink.

Although this construction of the knot treats its two loops differently from each other, the two loops are topologically symmetric: it is possible to deform the same link into a drawing of the same type in which the loop that was drawn as a figure eight is circular and vice versa.[2] Alternatively, there exist realizations of this knot in three dimensions in which the two loops can be taken to each other by a geometric symmetry of the realization.[1]

In braid theory notation, the link is written

Its Jones polynomial is

This polynomial and are the two factors of the Jones polynomial of the L10a140 link. Notably, is the Jones polynomial for the mirror image of a link having Jones polynomial .

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).[3]

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.

History

Old Thor's hammer archaeological artefact

The Whitehead link is named for J. H. C. Whitehead, who spent much of the 1930s looking for a proof of the Poincaré conjecture. In 1934, he used the link as part of his construction of the now-named Whitehead manifold, which refuted his previous purported proof of the conjecture.[4]

See also

References

  1. ^ a b Skopenkov, A. (2020), "Fig. 22: Isotopy of the Whitehead link", A user's guide to basic knot and link theory, p. 17, arXiv:2001.01472v1
  2. ^ Cundy, H. Martyn; Rollett, A.P. (1961), Mathematical models (2nd ed.), Oxford: Clarendon Press, p. 59, MR 0124167
  3. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, arXiv:0804.0043, doi:10.1090/S0002-9939-10-10364-5, MR 2661571
  4. ^ Gordon, C. McA. (1999), "3-dimensional topology up to 1960" (PDF), in James, I. M. (ed.), History of Topology, Amsterdam: North-Holland, pp. 449–489, doi:10.1016/B978-044482375-5/50016-X, MR 1674921; see p. 480

External links

This page was last edited on 26 December 2021, at 18:56
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.