Whitehead link  

Braid length  5 
Braid no.  3 
Crossing no.  5 
Hyperbolic volume  3.663862377 
Linking no.  0 
Unknotting no.  2 
Conway notation  [212] 
AB notation  52 1 
Thistlethwaite  L5a1 
Last /Next  L4a1 / L6a1 
Other  
alternating 
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 nownamed Whitehead manifold, which refuted his previous purported proof of the conjecture.
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Transcription
Contents
Structure
The link is created with two projections of the unknot: one circular loop and one figure eightshaped 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
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 twocusped 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 figureeight knot, and Dehn filling on both components can produce the Weeks manifold, respectively one of the minimumvolume hyperbolic manifolds with one cusp and the minimumvolume hyperbolic manifold with no cusps.
See also
References
 ^ Agol, Ian (2010), "The minimal volume orientable hyperbolic 2cusped 3manifolds", Proceedings of the American Mathematical Society, 138 (10): 3723–3732, MR 2661571, doi:10.1090/S0002993910103645.