Whitehead link

Whitehead link
Crossing no.	5
Linking no.	0
Unknotting no.	2
A–B notation	52 1
Thistlethwaite	L4a1

Simple depiction

Old Thor's hammer archaeological artefact

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

J.H.C. 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 Marilyn's Cross. (Note that $V(1/t)$ is the Jones polynomial for the mirror image of a link having Jones polynomial $V(t)$.)

References

Weisstein, Eric W. "Whitehead link". MathWorld.

See also

• Solomon's knot
• Weeks manifold

External links

• L5a1 knot-theoretic link on Knot atlas site