Whitehead link

Whitehead link
Braid length	5
Braid no.	3
Crossing no.	5
Hyperbolic volume	3.663862377
Linking no.	0
Unknotting no.	2
A-B notation	52 1
Thistlethwaite	L5a1
Last /Next	L4a1 / L6a1
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

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

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

Its Jones polynomial is

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

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

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.

See also

• Solomon's knot
• Weeks manifold
• Whitehead double

References

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

• "L5a1 knot-theoretic link", The Knot Atlas.
• Weisstein, Eric W. "Whitehead link". MathWorld.