Whitehead link

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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.

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The link is created with two projections of the unknot: one circular loop and one figure eight-shaped (i.e., a loop with a Reidemeister Type I move applied) loop 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.

In braid theory notation, the link is written

:$sigma^2\_1sigma^2\_2sigma^\{-1\}\_1sigma^\{-2\}\_2.,$

Its Jones polynomial is

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

References