Whitehead link

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Whitehead link
Whiteheadlink.png
Braid length 5
Braid no. 3
Crossing no. 5
Hyperbolic volume 3.663862377
Linking no. 0
Unknotting no. 2
Conway notation [212]
A-B notation 52
1
Thistlethwaite L5a1
Last /Next L4a1L6a1
Other
alternating


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[edit]

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

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[edit]

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.

See also[edit]

References[edit]

  1. ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2-cusped 3-manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, doi:10.1090/S0002-9939-10-10364-5, MR 2661571 .

External links[edit]